Fermat’s Conjecture

My greatest passion is for math – not engineering, physics, computer science, programming, project management, nor – even – astronomy. No, math, often called the Queen of the Sciences, is my passionate love affair, my secret mistress, and a lasting infatuation of mine. Like most good love stories, the object of my affection is often beyond my grasp. Yet I continue to pursue her, flowers and chocolates in hand, as I chase her from café to café.

It was not always so, astronomy was my first love, followed by physics and electronics, and even geology caught my fancy at one time. But math, pure thought stuff, yet much more disciplined than programming (even though programming is of the mind too) that I keep coming back to time and time again.

“If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.” – John von Neumann.

One of the happiest times in my career was when I wrote two textbooks for use in IBM’s Programmer Retraining Program. I took about eight or ten months to focus on the creation of the texts. I had an intern to assist me, and I spent days at a terminal writing. I taught myself IBM’s Generalized Markup Language or GML, which was marketed as “IBM Script.” (I did take a one-week class on Script part way through the writing sessions.) Script had a mathematics markup extension, and I did all the editing on a mainframe computer. The mathematics book I wrote was a self-study manual on Discrete Mathematics, the branch of math associated with programming. I called the book “Logical Expressions.”

I also wrote a short tutorial on the use of IBM’s Personal Editor that contained some basic training on using the IBM PC. The students in Programming Fundamentals used these two textbooks in their training. The math textbook served as a self-study guide, and it was used before students took a preliminary test to qualify for the retraining program. I also taught a two-week-long math course based on my book in Boulder. I think IBM published around 5,000 copies of both texts, so they weren’t exactly best sellers, but it was a very joyous time for me to be immersed in math and writing. It may have been the genesis of what I’m doing right now!

It was a time of deep focus and daily successes. I loved to see my creation growing in the drafts produced by the printer, and, day-by-day, I added to the body of work that ultimately was read by hundreds, possibly thousands of students. They may not have shared the joy I felt in creation of the books, but they – at least – read them. This adventure increased my understanding of writing and printing and even introduced me to Donald E. Knuth, one of my long-distance mentors.

I’ve written about math before, and discussed the beauty of equations and described the deep connection I have with Maxwell’s Equations, the actual formulas that describe light and all electromagnetic radiation. However, to even read one of these fundamental equations requires a very advanced knowledge of math. Maxwell’s equations use Differential Calculus and a special short hand called Div, Grad, and Curl. I wrote about my discovery of these powerful mathematical tools before and how they influenced my thinking. Here are several links to those stories.

The Most Beautiful Formula in Mathematical Thought

A Treatise on Light (and Color)

A Note on Mathematics and the Importance of Gödel’s Theorem

A Monograph on the Special Number 36 and the Fundamental Theorem of Arithmetic

How to explain mathematics – especially to non-mathematicians, and that includes most out there? Sure, you all know about arithmetic, and even a smattering of algebra, geometry, trigonometry, and maybe even some calculus. When does the beauty appear? How much must you know to see that beauty? That is the question.

What if I told you about one of the hardest problems in all of mathematics? Would you say, “I can’t even understand the questions, much less the answer.”

This won’t be hard. It just involves some simple formulas with exponentiation … powers. Not hard. You know: squares and cubes, etc., plus some addition. It’s easy. Yet this problem has troubled mathematicians since it was first stated … and that started several thousand years ago.

The problem is very easy math. The solution turned out to be very difficult math; very difficult indeed.

A brilliant mathematician in the seventeenth century wrote in a margin of a mathematics book:

It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second, into two like powers. I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.

The unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century. It is among the most famous theorems in the history of mathematics and prior to its 1995 proof was in the Guinness Book of World Records for "most difficult mathematical problems".

So this is a tough problem … very tough … yet very beautiful in its simplicity. You don’t need advanced math at all to understand the problem. The solution … on the other hand … is very, very deep. From this simple problem, so much as been developed.

This problem involves Pythagorean Triples. You all remember ancient Pythagoras and his famous theorem regarding right triangles. This is related … and just as old.

Here’s the equation:

a² + b² = c²

This is also called “splitting a square.” “Split a given integer square into the sum of two integer squares.” It turns out that there are many, many … possibly an infinite number … of values for a, b, and c that will make the equation true.  The simplest, 3, 4, and 5.

3² + 4² = 5²,  9 + 16 = 25

When applied to a right triangle, with two sides equal to 3 and 4, then the hypotenuse will be 5. You remember, the square of the hypotenuse equals the sum of the squares of the other two sides. This is the famous “3, 4, 5 triangle” that demonstrates this theorem.

The Pythagorean triples must be integers. That doesn’t seem to limit things much. There are many, many Pythagorean Triples ... as I mentioned previously. Try 5, 12, and 13. Find some others.

This basic math goes back even further than the Greeks. We know the Babylonians knew of this theorem and the triples and the Egyptians used this formula to build the pyramids and make them “square.”

The ancient Chinese and Indian mathematicians knew these formulas too, and they used them to create right triangles for surveying, carpentry, masonry, and construction. It is very useful, practical math.

The Greeks were the ones that focused on the beauty of the math and its possibilities. There certainly is beauty in using the three integers, 3, 4, and 5, to create a right angle. What simple and powerful ideas … so often created and recreated by the mathematicians of many early cultures.

This leads us to the realm of Diphantian Equations. A Diophantine equation is a polynomial equation in which the solutions must be integers. Their name derives from the 3rd-century Alexandrian mathematician, Diophantus, who developed methods for their solution. A typical Diophantine problem is to find two integers x and y such that their sum, and the sum of their squares, equal two given numbers A and B, respectively.

It is our old friend a² + b² = c².

Now, as I’ve already spoken at length, there are many solutions to this equation. Many values of a, b, and c that are integers (whole numbers) and that make the equation true.

So let’s add another variable. Consider:

aⁿ + bⁿ = cⁿ

This is a Diphantian equation, a, b, and c, as well as n must be integers. Further, as we’ve already discussed, if n = 2, there are a lot of solutions for a, b, and c. But what if n = 3?

What integer solutions are there to a³ + b³ = c³? Well, no one has ever found such a, b, and c.

Same for n = 4, or 5 or 13, or any other number. There were no solutions for n > 2. Now, how to prove this? (This part here might require some mathematical thinking. Even though, for hundreds or thousands of years, no one had found solutions for n > 2, you can’t be sure there isn’t one, just, as yet, not discovered. A mathematical proof, on the other hand, would end the search. If it could be proven that no solution exists for any n > 2, that would be the end of the quest.)

Then, along came Fermat. That’s Pierre de Fermat, born around 1601 and died in 1665. He was a lawyer in Toulouse, France. He was also an amateur mathematician who is given credit for early developments that led to the infinitesimal calculus, including his technique of adequality. In particular, he is recognized for his discovery of an original method of finding the greatest and the smallest ordinates of curved lines, which is analogous to that of the differential calculus, then unknown, and his research into number theory. He made notable contributions to analytic geometry, probability, and optics. He is best known for “Fermat’s Last Theorem,” which he described in a note at the margin of a copy of Diophantus’ “Arithmetica.”

It is actually more correct to call it “Fermat’s Conjecture,” since he did not publish a proof. That was the name used for most of the history of this conundrum. Recently it has become known as “Fermat’s Last Theorem,” and that is the term most likely to be familiar to non-mathematicians.

He communicated most of his work in letters to friends, often with little or no proof of his theorems. This allowed him to preserve his status as an "amateur" while gaining the recognition he desired. This naturally led to priority disputes with contemporaries such as Descartes and Wallis. He developed a close relationship with Blaise Pascal. These may not be household names, but they are some of the founding fathers of modern math.

He was reading a book on Diphantian mathematics and the particular problem I’ve been discussing. In the margin of the book he wrote, “I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.” This became known as “Fermat’s Conjecture,” and later “Fermat’s Last Theorem.” He never provided his proof, although he did provide a proof that is equivalent to proving that solutions with n = 4 would be impossible. He used a method called “infinite descent” to prove that case. But there was no record of his, complete, “marvelous proof.”

Now he was not a braggart, nor a mathematical lightweight. So his statement was taken very seriously.  Together with René Descartes, Fermat was one of the two leading mathematicians of the first half of the 17th century. Was he wrong?

(A reminder that Descartes discovered the Cartesian Coordinate System, that familiar graph of x and y that is used to merge algebra and geometry into Analytical Geometry, a forerunner of the Calculus. Descartes established the basic concepts of Calculus, but didn’t quite make it all the way. That came later when Sir Isaac Newton and Gottfried Wilhelm Leibniz independently finished the job.)

Lacking Fermat’s proof, over the next two centuries (1637–1839), the conjecture was proven for only the primes 3, 5, and 7, although the great female mathematician Sophie Germain proved a special case for all primes less than 100.

(Marie-Sophie Germain (1776 – 1831) was a French mathematician, physicist, and philosopher. Because of prejudice against her gender, she was unable to make a career out of mathematics, but she worked independently throughout her life.)

The list of mathematicians that contributed proofs for certain classes of numbers reads like a who’s who of important mathematicians including Euler, Legendre, and even Carl Friedrich Gauss, a man I consider the greatest mathematician of all time.

In the mid-19th century, Ernst Kummer proved the theorem for regular primes. Building on Kummer's work and using sophisticated computer studies, other mathematicians in the 20^th century were able to prove the conjecture for all odd primes up to four million. But these were all limited proofs of what is called “special cases.”

The final proof of the conjecture for all n came in the late last century. In 1984, Gerhard Frey suggested the approach of proving the conjecture through a proof of the modularity theorem for elliptic curves. Building on work of Ken Ribet, Andrew Wiles succeeded in proving enough of the modularity theorem to prove Fermat's Last Theorem, with the assistance of Richard Taylor. Wiles' achievement was reported widely in the popular press, and has been popularized in books and television programs.

So, finally, in the final years of the twentieth century, a final proof was found and offered and reviewed. It is very complex, using powerful methods, some suggested by other work of Fermat’s. British mathematician Andrew Wiles and his former student, Richard Taylor, finished a long work on the problem and published two papers describing the proof in the Annals of Mathematics. It was 358 years after Fermat wrote in the margin of a book.

Fermat’s Conjecture was not just a puzzle to be solved. Like most of the progress of mathematics, it led to new discoveries, which led to more discoveries, which led to the overall advancement of the understanding of the deep mysteries locked in numbers and equations.

Still an open question is if Fermat really had the answer. Most doubt he had a complete answer. The final proof used many advanced methods that simply didn’t exist in Fermat’s day. It is more likely that he had solved the problem for a subset of all possible exponents. Recall we have his solution for n = 4. Perhaps he had other solutions in addition or a method to generalize that solution. But most modern mathematicians don’t believe he had a complete answer. But we will never know for sure.

Even if he hadn’t found the marvelous proof, Fermat’s Conjecture is considered one of the greatest mathematical problems of all time. That it took so long to solve is a testament to the energy of mathematicians through the ages. The problem seems so simple, and yet the solution so hard to produce.

It is a sort of revenge for all math students who struggled with the problems in textbooks. Nice to know even the authors of the textbooks had problems that they struggled with too. Even nerds can encounter unsolvable problems. Sort of humbling … if you are a nerd.

I find beauty in the deep thoughts, the powerful arguments, and creativity of the solutions, and the pure difficulty of these problems. I see the history of all struggles in the tales of these solutions. Even in science we see prejudice, but truth wins out ultimately. Like a runner who finishes well in a race, solving these problems is a thrill … if you can solve them. And studying the solution and the path taken to those solutions can be most uplifting.

On Primes

Today is the start of 37 years that Linda and I have been together. Actually we've been together longer than that, but the 37th year of marriage begins today.

 Yesterday I wrote about the number 36 and primes and factoring. Thirty-seven can't be factored. It is a prime number. The only multipliers are one and thirty-seven. Recall a prime has no divisors except itself and 1. So 1 x 37 is the only way to get 37. Next year's 38 is not prime. No even number, except for 2, is prime. After all, the definition of an even number is a number divisible by 2. Thirty-nine isn’t a prime either, it’s 3 x 13. Forty is no prime. The next prime in my anniversary count-up is 41.

 There are patterns in prime numbers. If you write out all the numbers in a big list, and then mark out the numbers that are not prime, you will see patterns. Of course, there are a lot of numbers … actually an infinite amount of just the natural numbers, the integers equal or greater than zero. To begin with, you can eliminate every other number … the even numbers. Besides, it is easy to tell if a number is even, even if it is a large number. Recall that the even numbers all end with 0, or 2, or 4, 6, or 8. You can eliminate a few more. Any number ending in 5 (except for 5 itself) can't be a prime because it is divisible by 5. All numbers ending in 0 are divisible by ten, but we already eliminated them as even numbers.

 So that means a prime number, even a very, very large prime number, must end in 1, 3, 7, or 9. Of course, just because a number ends in one of these doesn't make it a prime. For example, 21 ends in 1, but isn't a prime; 21 = 3 x 7. Same with the other numbers I listed. I'll leave finding a counterexample, that is a number ending in 3, 5, 7, or 9 that isn't a prime, as an exercise for the reader. Note you cannot prove a theorem true by example, but you can prove it false.

  
 Determining if a number is prime is more complicated than just examining the last digit. With very large numbers, it can be quite difficult to determine if they are prime or not. But all candidates to be prime must end in those four digits.

 (By the way, 13 is a prime. So is 23, 43, 53, 73, 83, and 103. But 33 is not. It is 3 x 11. Nor is 63 = 3 x 21 or 93 = 3 x 31. Guess the next nonprime ending in 3. Are you starting to see the patterns?)

 There is a deep and rich and ancient and practical use for all this analysis of primes. It is an area of math called "Number Theory." The ancient Greeks originally studied it nearly a thousand years before Christ was born. All through history, number theory has interested some of the greatest mathematicians of all time and a few very talented amateurs. To this day there are unproven conjectures in number theory and, as I've mentioned previously, it is a fundamental foundation to modern cryptography used to make the Internet safe, provide secure communications, and even implement "digital signatures."

 An ancient method of studying the patterns of primes, at least for the first few hundred numbers, was developed long, long ago. It is still used as the basis for determining primes with very large numbers. The "Sieve of Eratosthenes" is an ancient idea still used in modern computer programs.

To find all the prime numbers less than or equal to a given integer n by Eratosthenes' method:

1. Create a list of consecutive integers from 2 to n: (2, 3, 4, ..., n).
   
   
2. Initially, let p equal 2, the first prime number.
   
   
3. Starting from p, count up in increments of p and mark each of these numbers greater than p itself in the list. These will be multiples of p: 2p, 3p, 4p, etc.; note that some of them may have already been marked.
   
   
4. Find the first number greater than p in the list that is not marked. If there was no such number, stop. Otherwise, let p now equal this number (which is the next prime), and repeat from step 3.

When the algorithm terminates, all the numbers in the list that are not marked are prime.

As a refinement, it is sufficient to mark the numbers in step 3 starting from p², as all the smaller multiples of p will have already been marked at that point. This means that the algorithm is allowed to terminate in step 4 when p² is greater than n.

Another refinement is to initially list odd numbers only, (3, 5, ..., n), and count up using an increment of 2p in step 3, thus marking only odd multiples of p greater than p itself. This actually appears in the original algorithm.

You can see why this is called a “sieve” because the nonprimes are sifted out.

There’s a lot more to number theory, and it includes a lot of fascinating games and some really deep insight into the mathematical building blocks. Some search for the face of God in these deep, natural insights. Certainly it seems the “numbers” were created by the Lord, not just invented by man; much more of a “discovery” than an “invention.”

There is much, much more to learn, but I’ll just list the first 1,000 primes and let you do a little exploration.

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997

I mentioned the use of primes in cryptography. Let’s explore that a little. What is needed is for a successful encryption is a “trap door function.” That’s a calculation that is relatively easy to perform, but almost impossible to determine the inverse function. That is, the calculation that would undo the original.

Consider how easy you could multiply two large prime numbers. (Of course, first you need to determine two large prime numbers. Use the “Sieve.”) Let’s take 911 and 857. Now they can easily be multiplied … even easier with a calculator or computer. 857 x 911 = 780,727.

Now for the inverse function: Given a large number that is known to be the multiple of two large primes, determine the two prime factors. Suppose you didn’t know 857 x 911 created 780,727. How would you determine the factors of 780,727? You might even think it could be a prime … it ends in 7. You could start by taking a list of primes and start dividing. You do have to start at the beginning because this could be 13 x 60,057 … (but it isn’t, 780,741 is).

You could solve this problem in an hour or so with a calculator and systematic testing. A computer program would solve it in seconds. OK. Make the numbers bigger. Suppose the two primes are each 64 bit numbers. Those are numbers can be as large as 9,223,372,036,854,775,807. That would take your personal computer months if not years to calculate. A very fast computer might get it in just hours.

However, if the primes used were as large as 256 bits, then, most likely, even the government’s fastest computers couldn’t solve it in less than years of running.

Yet, if you knew one of the factors, you could divide and calculate the other factor in just seconds … or minutes if doing long division by hand.

That’s the magic trap door function. If you use a number that is the multiple of two large primes as an encryption key, no one can guess or “reverse engineer it” easily or … more significantly … quickly! However, if you know one of the prime factors, you can quickly verify the number and determine the other prime.

I won’t go into any more detail as it gets complicated and all mathy and programmy, but you now see the basic idea. Prime numbers and the difficulty of factoring is at the heart of the method.

Wait, how do we know that some smart mathematician hasn’t figured out a shortcut method to factor the large number quickly. Well, we don’t know. But we do know that mathematicians since ancient times have been looking for that algorithm, and apparently have not found it … yet.

So number theory isn’t just for the ivory towers. IBM and the NBS and the NSA and many other interested parties have worked for years to perfect the methods of encryption … all based on the difficulty of factoring large numbers that are the product of two primes.

It was even in the news yesterday that a new algorithm had been found to use for encryption after five years of research. They chose to use an old method known for many years. All that “state-of-the-art” research and we’re back to the ancient ways. That’s why we think these methods are secure.

Do a little “Google” exploration and discover more. Wiki’s an excellent source.

A Monograph on the Special Number 36 and the Fundamental Theorem of Arithmetic

A monograph is specialist work of writing on a single subject or an aspect of a subject, usually by a single author. Well, I’m not single … in fact … today, I’ve been married for 36 years. That alone makes 36 special.

Now most non-mathematicians focus on “decades.” For example, a tenth anniversary, a fiftieth anniversary, a centennial. And that is fine. Some like the “half-decade,” such as a twenty-fifth anniversary. Those are meaningful milestones. Why, just last year, we had our 35th wedding anniversary. That’s the Coral or Jade Anniversary (US is confused between the two, in the UK it is just Jade). I didn’t get her Jade or Coral, but I did get her Tourmarine earrings (with lots of diamonds … always get those ladies lots of diamonds.)

But, as an integer, 35 is rather boring. It belongs in the class of numbers that have only two prime factors. Five time seven. That’s it.

The Fundamental Theorem of Arithmetic, also called the "unique factorization theorem" or the "unique-prime-factorization theorem," states that every integer greater than 1 is either prime itself or is the product of prime numbers, and that, although the order of the primes in the second case is arbitrary, the primes themselves are not.

Stated a little less technically, any integer can be factored into prime factors in one and only one way. (Order of factors doesn’t matter.) By the way, integers are numbers that can be written without a fractional or decimal component. For example

…, -2, -1, 0, 1, 2, 3, …

and prime numbers are a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number is an integer greater than zero, or

1, 2, 3, …

I’m not going to define divisors. This is turning into an infinite series.

So, let’s examine (but definitely not prove … you don’t prove by example) the Fundamental Theorem of Arithmetic.

For example, 36 can be factored as 6 times 6. And 6 is 2 x 3. So 36 = 2 x 2 x 3 x 3. (Remember, order doesn’t matter in multiplication.)

But 36 also is 2 x 18. And 18 = 2 x 9. And 9 = 3 x 3. So that yields 2 x 2 x 3 x 3. The same thing.

Wait, isn’t 36 = 3 x 12? Why yes it is. And 12 = 3 x 4, and 4 = 2 x 2, therefore 2 x 2 x 3 x 3.

So, whether you start with 2 x 18 or 3 x 12 or 4 x 9 or 6 x 6, you always end up at the basic factorization of 2 x 2 x 3 x 3.

Now 1 is not considered a prime factor. It would just “get in the way” of the math because one times any number is the original number, and any number can be divided by one … actually over and over and over again. 1 x 1 x 1 x 1 x N = N. No change; so that sorta takes the fun out of one.

Still, you can consider 36 = 1 x 1 x 2 x 2 x 3 x 3. In the words of the church lady, “Isn’t that special.” By the way, that's 1² x 2² x 3². Really special!

Prime integers aren’t just mathematician’s wet dreams. Why the whole system of computer encryption is based on the difficulty of factoring large numbers made up of only two primes. Without that encryption, there would be no secret messages nor secure computer communications. So primes … and the very fact they’ve been studied since ancient (Greek) times … is at the fundamental foundation of our modern digital society.

Yes, 36 is a very special number. And my wife Linda is a very, very, very special lady. I love her this much = 10 raised to the 100 power, raised to the 100 power, raised to the 100 power. That’s a hypergoogleplex. No, on second thought, I love her a lot more than that.

(Let's see if html can handle a hypergoogleplex: 10^100100100. Wow … I think I love html as much as I love math.)

Why I love her "This Much" … holding out arms like a fisherman explaining a large fish … assume length of left arm is minus infinity and length of right arm is positive infinity. Now we're cooking.

Of course, infinity isn't a number, it's more like a "limit." Oh great, now I'm going to have to explain "limits." This will take a while. Better sit down. … (to be continued?!?)

Happy Anniversary love of my life = LRC.

Finally Falling oFF the Fiscal cliFF! OH F… !!!

I’ve written about his little adventure before. I first warned about it in the article “Taxmageddon” originally written Feb. 20^th of this year.

Taxmageddon

I added another warning with “Taxes” on July 24 to explain tax brackets and what that light at the end of the tunnel was … an oncoming train.

Taxes

Bad things come in threes, so I followed that with this article on Dec. 7, when I predicted that the lame duck Congress would not take any action.

Don't Worry

Of course, there’s still time. Anyone really think Congress will act today or Monday?

No, ain’t gonna happen!

Still, it isn’t really a cliff. This set of tax increases and spending cuts that go into automatic effect on Tuesday, January 1, 2013 will have more of a gradual impact throughout the months of 2013 and won’t have a final splat until April 15, 2014. There is still time for Congress and the President to act in January, or February, or March, or even next December.

The effects will be gradual. In fact, due to uncertainties, the IRS hasn’t even published updated withholding tables yet, so pay on Jan. 1 won’t change. Let’s talk a little about how income taxes are collected in this country.

For people who work for other people, they pay taxes via “payroll withholding.” Based on your salary and the number of exemptions and something called the “IRS Withholding Tables,” your employer will hold out some amount of your paycheck and send it to Uncle Sam. If you’re self-employed, the process is similar, only you send the money once every three months.

Then, at the end of the year … usually by the very end of January, the next year … you get all these financial forms. W-2s and 1099s and other statements on taxes paid, interest earned, income, real estate taxes, etc., etc., etc. Then it is a race to April 15 to complete your taxes and determine how much you owe for the previous year. Hopefully, and typically, people have already paid more than they owe through the withholding. That means you will soon receive the wonderful “Tax Refund Check” that usually means bills get paid or a new commercial item will soon grace your home: 60” TELEVISION for EVERYONE. Of course, some didn't withhold enough. Better get out the checkbook.

So, since the new tables aren’t out yet, and won’t be for several weeks assuming Congress doesn’t act, and it will take your employer anywhere from one to two months to get the tables installed in their payroll systems, we may not see any income tax increase until March. Of course, then the systems will actually collect more money to make up for the shortfall.

How much your income tax goes up depends on many factors … most importantly your income. Average will be in the neighborhood of around $2,000 for the year. That will equate to around $100 less in your paycheck, assuming you get paid twice a month.

Then there’s the famous 47%. Those are the people that don’t pay income taxes. Maybe they don’t receive a paycheck. We still have very high unemployment is the US. Most of the 47%, however, don’t pay any income taxes because their income is too low, especially if they have a large family. Now I’m not offering to trade places with those folks. It is tough to make ends meet on a minimum wage income. At least those poor people don’t have to pay income tax. But … guess what? … anyone earning a wage pays Medicare and Social Security taxes. Oh, wait, the rich don’t pay SSN, at least on income above $113,700 next year … but I digress.

Well, for the last two years, all wage earners have enjoyed a 2% cut in the 15.3% FICA tax. (That includes your employers share … you directly pay 7.65% unless your self-employed, in which case you pay the whole enchilada.) That tax cut also ends this year, and, I suspect, on Jan. 15, when you get your first paycheck of 2013, you’ll see that … those tables are already programmed into the payroll computer.

There is one final increase in January. That’s the taxes on capital gains. Sure, that’s just for rich people … or anyone who sells some property … or anyone selling stocks and bonds they saved for retirement … or anyone … you get the idea. That increase also “depends,” but, in general, it will be a 5% increase to a high of 20%.

Heck, there may even be problems with the AMT or Alternative Minimum Tax. That’s a second calculation of income tax enacted to catch rich people that had tons of deductions … they had to, at least, pay a certain amount of tax. Unfortunately, with inflation, the AMT rates now hit middle class persons, especially those in high state tax states. (State taxes are deductible … you know!) So, in typical “kick the can down the road” fashion, congress has been adjusting the AMT annually. Well, guess what? … again! … that adjustment is about to expire too.

Then there's the end to the end of the marriage penalty, and a change in inheritance tax (also called DEATH TAXES). Now don't take my word for it, I'm not sure about all these things and figures, but it is going to be one high cliff that we meet on JanOne.

So, I can’t tell you how much your taxes are going up next year, but I guarantee they are going up, probably around 5% minimum. So, if you make $30,000 a year, that’s $1,500 bucks … make $60,000, probably $3,000. Make more? … well, “There’s one for you, nineteen for me.” Oh, sorry, that’s England. But we’re working on that here in the US.

So, you will have less money to spend next year … and you … and you … and ME!!

So, tighten your belts.

Wait, there's more: we have a “consumer driven economy.” Less spending means less economy which means less purchases which means less manufacturing and sales commisions which means less employment which means even lower national income which means more less spending and more less purchasing and more less manufacturing and more less employment. Why, it’s a vicious cycle. Where will it end?

It will end in a recession … that’s where. Oh, wait, there’s still government spending. That’s help. After all, Uncle Sam still has his credit card. Uncle will save us. He’ll just spend the hell out of the Treasury with some kind of stimulus or bailout or rescue … at least we’ll all get food stamps.

But wait … there’s more. The last time Congress addressed the “Debt Ceiling” -- that’s a limit on borrowing -- the only way the budget hawks would raise the ceiling is if Congress promises to lower spending. They had a plan. I won’t get into details. It didn’t happen. So now AUTOMATIC CUTS are coming. Seems Congress promised to lower spending and put that into the law. That law comes into effect next year too.

Frankly I’m too tired to calculate how much. Let’s just assume 5-10% cut in spending across the board. Some things will be cut: military, discretionary. Some won’t be cut ... Social Security … at least not yet. But they are big cuts and they will also reduce our economy further.

Here are some numbers. Right now, most of the world is enjoying 0% economic growth … sort of enjoying like a toothache. Heard of Greece? Heard of Spain? Or even England? You really should listen to the radio more. Or read a newspaper. Or read my blog!

In our happy home, the United States of Credit Cards, we are enjoying about 1.5% annual growth. Calculate in a 5% raise in taxes and a 5% or more cut in government spending. Go ahead. Just subtract it from the 1.5% growth. It’s more complicated than that, but what the hey. So, what does your calculator say? NEGATIVE NUMBER. That can’t be good. Negative economic growth … why that’s called a RECESSION or, if bad enough a DEPRESSION. Yup, it is depressing.

So now you know. When the Fiscal Cliff meets the Debt Ceiling … we’re in trouble. Oh, one more thing. We need to vote for a new debt ceiling as soon as Feb. But, I’m not worried. Surely Congress will agree and raise the ceiling.

No, my name isn’t Surely, and I don’t think they will. I think we’ll default on our debt, at least for a short time. Never mind Taxmageddon … it may be Financialmageddon…a rum dum.

It won’t happen all at once, but gradually. Know the real problem of all this uncertainty? Businesses are sitting around waiting for the dust to settle. Businesses don’t like uncertainty. If they’re not sure, they won’t spend. And business spending equals jobs. Batten down the hatches. We’re in for a blow. Mixed metaphors sighted off the port bow. The Fiscal Cliff is hitting the Debt Ceiling and we’ll end up burning dollar bills to keep warm. YIKES!!

Unfinished Dreams

Personally, I feel like I was very successful in life. Of course, there are a few more years to go before I finish my race, but now that I’m retired it affords me time to reflect over the first 65 years of my life, my career, my marriage, my family, and just how well things went for me. Let me start by stating I’m particularly blessed. I can’t imagine all the good things that happened to me in my lifetime as being real ... it has to be a dream! Starting with wonderful parents who raised me in a wonderful little town full of happiness and childhood dreams. That was the perfect beginning. I got to do most all the things I wanted to do as a child, and got support from all those that loved me.

As a family, we traveled and I got to enjoy the beauty of the Lord’s creation in my home state of Montana and around the country. We picnicked and camped and boated and fished … I didn’t hunt. My love of beauty and nature grew out of that childhood.

I worked more than I cared to at the time, but that employment gave me valuable life lessons as well as a good income growing up. I got to drive the delivery truck and that was part of my on-going love affair with things with wheels.

I sputtered a bit on the education front, flunking out of the Montana School of Mines, but I did gain valuable guitar skills and party skills, and met some more great friends. I was a bit lonely being away from home for the first time, but that was a good life lesson too.

I spent some more good times with friends, partly aided by a broken arm that gave me a draft deferment for over a year. I lost a good friend in an automobile accident, and that may have been the only real grief in my young life. I worked in mine, mill, and smelter; and was no stranger to manual labor. At night and weekends, the music played, the girls danced, the friends gathered round, and happy times were had by all.

Eventually I ended up in the Navy, enlisting for six years to gain advanced electronics training. That was a childhood dream that I finally accomplished. I spent my military career as part of the Navy’s Atlantic Fleet, thereby avoiding the conflict in SE Asia. That was probably a blessing in disguise. I made great and wonderful friends during that time and engaged in motorcycles and mechanics and lived independently. It was a time of music and parties, and I partook of those with great vigor.

After finishing my national service (with one humongous two day party on the beach), I spent almost a year in Spokane, Washington living with my parents. I had no responsibilities, and I learned to sleep late. I got my First Class Commercial FCC License, and did some work in both radio and television before embarking on the next leg of my life’s journey. My original goal was to work for a large music store in downtown Spokane as a musical instrument repairman, but that dream evaporated. The radio and TV work substituted, and the FCC license turned out to be a key to another dream in another state.

I moved to Colorado in January, 1974. That had been a dream I shared with my shipmates, and many had planned to join me there. I had attended an Air Force School in Denver as part of my Navy training, and always dreamed I’d return. Many of my colleagues in the calibration lab on the USS Vulcan shared that dream, and I had visits from many shipmates during the first five years I lived in Denver. My best buddy, Woody, came out west and lived with me for nearly a year before returning to his ancestral home in New England.

Woody and I shared a great joy of motorcycle racing as well as a fondness for running. We would run five miles a day, and race bikes on weekends. I was also good friends with the Lincoln family. I met Linda’s first husband, Tom, on board the Vulcan. We got together later in Longmont, and I shared interests in both guitar and motorcycles with Linda’s brother Chuck. I knew Linda’s mom and dad, and helped a bit around their house. One time Chuck and I caught a motorcycle part on fire in his mom’s oven … long story!

At that time, I was working for A.R.F. Products as an electronics technician designing and testing advanced radio controllers for missiles and rockets … including the space shuttle, and Woody was seeking work. He had a degree in math and wanted to be a math teacher. The employment agency gave him a lead to the Electronics Technical Institute, but he didn’t want to teach electronics. I did. I applied. I got the job. It was my First Class FCC License that sold me to them. They needed someone to teach FCC License Preparation, and there I was. My long-held dream to be a teacher was fulfilled.

At the same time, I was back in school studying Electronics Engineering at what was called Metropolitan State College. (Now it is called Metropolitan State University of Denver … I’m not the only one who has changed.) It was an unusual school with no campus. The classes were held in various office buildings throughout Denver. That worked for me since I was living and working in Denver. The GI bill was quite generous in those days, and/or tuition wasn’t that high back then, and I got several scholarships, so I actually got more money from the government to go to school than it cost. I was working full-time, so money was not the problem it often is to a poor student.

Metro gave me a ton of credit for my Navy electronics, and I started getting straight-A’s in school. It wasn’t that hard since I taught electronics at ETI and then studied it at night at Metro. I ended up graduating with only one B, and that was in a one-credit class. So my final GPA was 3.97. Pretty close to 4.0.

In the meantime, Linda had divorced her first husband; we started dating; I fell head over heels in love with the sweetest women I’ve ever met … plus the prettiest girl that would ever even say ‘hi’ to me, much less date me, and even give me a kiss now and then; and we got married. That was my greatest dream fulfilled, and has been the anchor of my life since. Two sons and several grandchildren later we approach our 36^th anniversary this week and I’m more in love with her today than any day before. That will be true tomorrow too. Trust me, I’m a mathematician. I understand infinite series. (These things are blessings to my life!)

By this time, I had married, purchased our first house in Longmont, been hired by IBM, and was expecting our second child. I still had some GI bill left, so I moved across the street from Metro (which, by then, had built a large campus in Denver which was shared between a junior college, Metro, and a branch of the University of Colorado) to the University of Colorado at Denver, and began working on a Master’s degree in mathematics with a minor in physics. Most of the classes were in Denver, which I commuted to for both work and school until I started at IBM. I also took some classes at the Boulder campus of CU. I didn’t do as well in graduate school. I actually got a lot of B’s and even a C. As the doll Barbie says, “Math is hard.”

Even though CU had a very good initial program of math starting with an excellent class on Mathematical Proofs to help students make the transition, I struggled for a long time with advanced math principles. Doubly concerning to me, at the time, was how easy it seemed to be for my classmates. At one point the instructor of Advanced Calculus took me aside and said I was in the wrong class. She said I should take “Advanced Calculus for Scientists and Engineers.” She thought it would suit me better with my engineering background. I said no, I wanted to be a mathematician and this was the class I wanted. She said, “OK, but you’re not doing well.”

That was when I discovered the local (Longmont) Public Library. I would go there to escape the mayhem of a house full of kids and a wife, I learned how to concentrate and study like I never had before. It took me almost a year to gain those skills and force my mind to work like it had never had to before. Electronics came easy because it was always incremental steps. I did Ham Radio and built electronics projects as a kid, and then studied in the Navy, which was good, but not exactly rigorous, then I taught electronics and took my degree. It was all incremental learning.

This math was different. Sure I had studied math before. I even took Differential Equations … rumored to be the toughest class in engineering school. No problems there. But now, as a math major, it was deep … very, very deep. The physics went better, and — after a year in the study cubicles at Longmont Public Library — I caught onto the math. That Abstract Algebra II class that I only got a C, was the last C. I didn’t get all A’s, but I did get my Master’s in Mathematics.

I remember after one final exam I visited the professor to see my grades. I had a 68 on the final exam. He told me that was OK, it was the second highest grade in the class. I started to leave, and he asked for the test back. I said I understood, he might use that test again. He said, “No, I just might have to prove that some people actually passed the class.” Maybe my fellow students weren’t doing so well after all. Things got better and I slowly caught on to what was going on. At one point we developed under-determined matrices for the new, at that time, cat scanners from GE. I started seeing how all this theory had a practical side too.

I became such a fixture at the Library that I ended up being appointed to the Longmont Library Board. A few years later I was elected Chairman of the Longmont Library Board. I spent seven years working with the Longmont Library, and I led the process to automate the library (install the catalog computer) and laid the ground work for building the new library. Alas, with the Internet, I don’t go there as much any more, but that’s a story for another note. On with my dreams …

Eventually, I earned my Master’s. That was part of my dream, but my main goal was a PhD: A “Doctor” of Philosophy. Know what? I wasn’t even sure what PhD: EE or Math or Physics or possibly Education. My job at IBM was going well, and after several years as an engineer and mathematician in the disk drive manufacturing area, I joined IBM Technical Education as a member of the Software Engineering and Computer Science department.

I started traveling and teaching all over the US and even some out of the country. It was a great fifteen years I spent in IBM Education, and — as is typical — I learned a lot more than the students. IBM sent me off to school at Harvard and Vanderbilt in Nashville. I attended on-line classes, more at CU and also math classes at CSU. But all these were short classes. I completed an internal IBM curriculum that was actually part of the IBM Education department I worked in. It was called the University Level Computer Science Program. It was a series of almost 30, one-week long classes taught by a pair of college professors. Each week-long class was taught at an off-site conference center, although — near the end of my attendance — many classes were on-site in Boulder (to save money). Each class was 40 hours, so it was equivalent to a college class, and the professors came from over 120 universities. This was IBM’s process to keep their programmer staff well educated. Many early IBM programmers did not have CompSci degrees since that was a new subject. Many were engineers (like me) or mathematicians (like me) or physists (you’re catching on), or musicians …

I both graduated from and administered the ULCS program for many years, and am personal friends with many of those professors to this day. Sometimes I would co-teach with some of the college instructors in other IBM classes I ran on Software Testing. During this time, I built my reputation throughout the IBM Corporation as a testing guru and quality expert.

Finally, the strain of continual travel to teach and administer classes forced me to seek another job at IBM. Besides, I still had my PhD dream unfulfilled. I left IBM Education and started work with the IBM Printing Systems Division in Boulder. All this time I had never left Boulder. My Education job was all over the country, but my office and home was always in Boulder.

I started out at PSD (Printing Systems Division) leading the effort to prepare our systems for Y2K, the great date change issue facing us at the start of the new millennium. After an award winning success with that project, I was put in charge of the PSD SW Testing Lab. I ran the lab that did the final testing on all the IBM printing software. During this time I started attending the University of Denver seeking a Master’s degree in Computer Science. Although I already had a Master’s in Math, I wasn’t ready to decide just what PhD to go for, and I wanted to get an official certification of my career change from electronics engineer to software developer.

The DU classes strengthened my skills in C++, Java, and UNIX, which fit perfectly with the product development going on in Boulder in my lab. I was the lead tester and managed a million dollar lab with billion dollar staff. I loved working with my fellow testers and the relationships were deep and rewarding. Plus, we really got the job done, lowering field defects by double digit percentages year after year.

I made many physical improvements to the lab, installing advanced benches and computers and organizing the systems for maximum efficiency. I developed a lot of signage to guide the use of the equipment and created databases and processes to manage the workflow and organize tasks. Working with my manager, we developed an advanced training program and even testers from other divisions would attend our curriculum. (Once a teacher, always a teacher.)

I won several monetary awards and, one time, IBM sent Linda and me off for a vacation and conference in Hawaii for a week. My title changed from Engineer to Project Manager, and I completed the rigorous IBM Certification process. I began a time of consulting with customers and assisting other IBM Project Managers on projects that were in trouble. Again my reputation spread beyond the Boulder lab and, once again, I was traveling all over the country.

That was important because I had been a Senior Engineer and Senior Project Manager at IBM for about fifteen years at this point. In many organizations, “Senior” is the top rung of the career ladder. If you want to go higher, either in pay or status, you had to become a “Manager.” That is what, at IBM, is called a “People Manager.” Now I had people working for me in the testing lab, but I didn’t manage their pay or career. I just told them what to do daily. Their People Manager had the personnel  responsibility.

Fortunately, IBM has a dual ladder for executive levels. You don’t have to become a first-line manager, second-line manager, executive, vice president. You could climb just as high as a technical, non-manager role. As a Senior Engineer/Project Manager, I was already at the same pay grade as a first-line manager. But more was to come.

My “first-line” manager, Mary Barez, began a campaign to have me promoted to “Senior Technical Staff Member” or “Band 10.” (Senior Engineer was “Band 9.”) The ultimate goal was for me to become a “Distinguished Engineer” or Band D.

IBM Distinguished Engineers (DEs) are executive-level technical leadership positions and are appointed for outstanding technical contributions and leadership. IBM Distinguished Engineers are corporate appointments.

The expected career path for technical distinction at IBM is not well known or well documented, but it is called the Technical Resource Program. It is similar to the Fast Track Executive Resource program and has the following path:

• Senior Technical Staff Member (STSM), Executive IT Architect, or Executive IT Specialist
• Distinguished Engineer (DE) 
•  IBM Fellow.

To progress to the DE level an individual would be expected to be at an Executive grade (STSM) and have other distinguishing factors (for example, prolific inventors or patent holders) as well as being globally recognized experts in their respective fields, contributing to their clients' success, and IBM's growth.

STSMs and DEs are integral members of their units' executive teams, demonstrating leadership to these units and across the company by consulting with management on technical and business strategies and their implementation. They often have operational responsibilities for large, complex technical projects, and may have line management responsibility as appropriate.

As STSM I was a member of a council that planned our development budgets and provided special reports to management on direction and technical strategy.

Career progression can be towards the more senior technical executive position IBM Fellow, but also beyond that to more senior executive business management roles building on their technical achievements.

The IBM Technical Community numbers over 200,000 people, including over 600 Distinguished Engineers and about 5,000 STSMs.

So STSM was the first step. But my manager could not promote me to that level. It would require that the Division Manager give that promotion. Mary set the groundwork for the promotion, filling out paperwork and contacting technical leads in other divisions to give testimonials of my skills. She ended up transferring before the promotion occurred.

My new manger was the Director of the entire Quality Organization, and he continued to lobby for my promotion. The final step was when I led the adoption and acceptance of ISO 2000 by PSD. I didn’t do it all, there was a team of over a dozen that led our division to successful completion and certification, and it was an important business step for our group of around 3,000 employees. It allowed us to bid on contracts and get sales we could not get previously. It was a massive undertaking that involved creating a Quality Management Process database (I used Lotus Notes) and documenting all our development processes in a specific format. I then led the training (always the teacher) of all the employees in preparation for the audit and validation. We passed and were rewarded the ISO certification.

That led to my promotion to STSM. Now I was one of only eleven technical leaders in all of PSD, and I was responsible for the technical processes of our entire quality organization and overall product quality. Included in this promotion was a nice pay raise and I celebrated with the purchase of a new sports car. Daddy done good!

I continued with the goal of DE. I never expected I could become an IBM Fellow, but DE would be the crowning cap in my career. I continued to lead the quality effort at PSD and my boss was soon promoted to the Vice President of Quality, and I was his staff person. He gave me the title of Technical Quality Leader (or was it Quality Technical Leader), and I focused on the overall quality of all our products. I added hardware to my previous software responsibilities and dug deep into the job.

I developed a new tool for managing quality that I called QSAT for Quality System Analysis and Tracking. Basit Mustafa, an intern and later programmer at IBM worked with me on the project for months and we developed the new tool and presented it to the management committee. I insisted to my manager that Basit do the presentation, since he had done all the coding. (I was the system Architect or designer). My boss objected, but I talked him into it, and it was a valuable experience for Basit. (The biggest concern was that Basit would not be brief. It was an important meeting, and this was just one item on the agenda. I figured Basit would be more terse and laconic than me!)

It gave Basit important recognition and it established me as a person to work for. I didn’t steal credit from those that did the technical work. My motto was, “There is no limit to what you can do, if you don’t care who gets the credit.” I went on to champion many development processes, establishing Orthogonal Defect Classification at PSD, and working with a group developing a similar system for “Problems.” (Problems are customer calls with issues. Not all the issues are defects. In fact, most are not. OPC was developed to manage and learn from problems.) I was in line to get a patent for this new work, but IBM sold PSD before the long patent process was completed. 

I was then assigned to IBM Research and traveled around the country helping other IBM organizations implement OPC. That further increased my exposure around the company ... again teaching was key to my advancement.

I was also very involved with the IBM Academy of Technology. This was another step. The Academy consists of the top three to four hundred technical leaders in IBM. Membership is by invitation from the Academy, and selection is not done by IBM management. The Academy has an organization of Affiliates. I was soon a member of an Affiliate when I served as the Chairman of the “Boulder Technical Vitality Council,” an inter-divisional organization covering all the technical people in IBM Boulder — not just Printing Systems, but all the five divisions on this 5,000 employee campus. We held an hour-long technical presentation every Tuesday in the cafeteria, and I was a common speaker. The Vitality Council was responsible for on-going education and technical employee retention. (Again, once a teacher, …)

I attend the Academy’s annual meetings and was hoping that I would soon be offered membership. My hope was either I would be promoted to DE and then offered member ship, or — more likely — offered membership and then promoted to DE.

Unfortunately, fate intervened. IBM Printing Systems was sold to Ricoh in 2007, and I was no longer an IBM employee. In the words of Maxwell Smart, “I was this close.” I don’t know if I ever would have made DE or been accepted into the Academy. It seemed I was on the right path and knew the right people … it really is who you know AND what you know … you know. PSD had two DE’s, and that was quite a few for a division with only 3,000 employees, which included about 1,000 engineers. I think I was on track and my manager was working for it, but it was not to be.

I’m sure I never would have become a Fellow. I know one Fellow, and she’s a lady. She was one of the inventors of JPEG and MPEG. That gives you an idea what it takes to be an IBM Fellow. I worked with her on several projects, including some mathematical verification of test completeness. Google “IBM Fellows” and check out the well known engineering names in that small community of technical excellence. No, my goal … my dream … was to become a DE. I did make it to STSM, which is a lot more than most IBM top engineers. But I just couldn’t go.all.the.way. (I miss Howard Cosell.)

So that’s one dream I’ll never make. How about the other dream? PhD?

Well, I’ve got some news on that. I’ve been attending Stanford University since the beginning of this year. I take online classes, and I’m studying hard. The next step is to take the Qualification Tests or “Quals.” My goal is to take them next January. Not this January … a year from now. Assuming I pass the Quals, there are still lots more to do. Research, a dissertation, I’d have to go to California and work at the campus. My time line is that it would take me about five to seven years, with one year completed so far. To tell the truth, I don’t think I will make it. Just the financial cost seems too high, and for what? Oh, just a dream I’ve always had!

The realist in me says, "I’m just taking some classes and tests to see how far I can get." The dreamer in me says, “ ‘Dr. Cheatham’ would really sound good.” If you’re a betting person, I recommend the realist. I really, honestly, don’t think I’ll go.all.the.way. I’m just going to do this until I get tired of studying. I wonder if my old cubical down at the Longmont Public Library is still available.

Meanwhile, since I believe dreams require a sound track, I’ll leave you with this title.