Several people have asked me why 137 is my favorite number. So let me tell you a few things about it, so you can understand why this number is so mysterious.
137 in Mathematics
The number 137 (Roman numeral CXXXVII) is fascinating for several reasons:
- it is the 33rd prime number;
- the next prime is 139, with which it forms a twin prime pair, and is therefore also a Chen prime;
- it is an Eisenstein prime, with a real part of the form 3n – 1 and no imaginary part;
- it is the fourth Stern prime, i.e. a prime that cannot be expressed as the sum of a smaller prime and twice a square number;
- together with 131, it is a sexy prime (their difference is 6);
- it is a Pythagorean prime, i.e. expressible in the form 4n + 1;
- it is a strong prime, i.e. greater than the arithmetic mean of its two neighboring primes;
- it is strictly non-palindromic, i.e. non-palindromic in any numeral system with base 2 < k < n – 2;
- its reciprocal is a repeating decimal with a palindromic period: 1/137 = 0.00729927007299270072992700…
137 in Jewish Mythology
- Zhe word kabbalah ( ק ב ל ה ) has the numerical value 137 in gematria.
- According to the Bible, Abraham was 137 years old when he died, as were Levi, Ishmael, and Amram.
137 in Physics
In Bohr’s atomic model, an atom with atomic number Z = 137 would have its innermost electron orbiting at nearly the speed of light; the existence of an element with Z = 138 is impossible according to the Bohr model (today, in theory, the possibility of creating superactinides with atomic numbers between 121 and 153 has been raised, but so far we have only managed to reach 122).
In reality, however, it isn’t 137 itself that is interesting, but 1/137. For a long time, this was thought to be the value of the fine-structure constant discovered by Arthur Eddington. Today we know that it is not exactly that, but rather 1/137.03599084. Back in 1916, when Sommerfeld was studying the relativistic shifts of spectral lines, it still appeared to be 1/137. And, quite surprisingly, it turned out to be a dimensionless quantity. This is particularly remarkable, since most of our physical constants usually depend on the system of units being used.
It is fairly obvious that we get different numbers for the speed of light depending on whether we measure distance in meters, miles, or Viennese inches. The fine-structure constant, however, has no units, which leads us to think that if a civilization anywhere in the universe engages in physics, no matter how they construct their system of measurement, they will find this constant to be the same.
Note: That sounds beautiful, but the latest results suggest that the fine-structure constant may in fact not be constant at all, but rather varies across space and time in the universe. If this turns out to be true, it would mean that the laws of physics are not the same everywhere—something that raises many exciting questions about our current understanding of the cosmos.
The constant itself essentially characterizes the strength of the electromagnetic interaction—the coupling constant between the electron and the photon. Its magnitude is the square of the ratio of the electron charge to the Planck charge; in other words, the square of the electron charge divided by the product of Planck’s constant and the speed of light:
By its very nature, it crops up everywhere in quantum physics, so it’s no wonder that it has long captured the imagination of physicists. It seemed especially mysterious back when we believed its value was exactly 1/137.
Stories about 137
Pauli, Heisenberg, and Feynman also spent a great deal of time pondering whether it was merely by chance that its value is what it is, and why it differs so significantly from 1. After all, physical interactions weaken according to powers of this constant. According to Feynman, it even has practical uses:
“If you ever get lost—say you board the wrong plane and end up in a different country than the one you intended—you don’t need to do anything more than grab a piece of paper, write 137 on it in big numbers, and hold it up. Sooner or later, a physicist will come by, notice it, and help you out.”
They say that Richard Feynman once remarked that every physicist should carry the number 137 on some personal belonging. Werner Heisenberg is also quoted as saying that if physicists ever manage to uncover the true meaning of 137, all the paradoxes of quantum mechanics will be resolved. Who knows?
It is said that after Wolfgang Pauli’s death he arrived in heaven, and in recognition of his work as a physicist, he was granted a personal audience with the Almighty.
— My son Pauli, said the Eternal, you may ask me anything, and I shall answer. What would you like to know?
— Why is alpha equal to one over one hundred and thirty-seven?
The Eternal smiled, conjured up a blackboard, took a piece of chalk, and began to write equations. After a few minutes, He sensing that His disciple was waving his hand excitedly behind Him. He turned around and His questioning look, Pauli pointed to an equation and burst out indignantly:
— Das ist falsch! (That is wrong!)
There is another story about Pauli. Toward the end of his life he was in poor health, lying in a hospital bed as he awaited a critical operation. One of his colleagues came to visit him shortly before he was wheeled into the operating room. As the visitor was leaving, Pauli called after him:
— On your way out, take a look at my room number! he asked.
On the outside of the door, the number was written: 137. As it turned out, this was to be his last operation—he did not live much longer afterward.
My Personal Connection to the Number 137
The fact is, I live at exactly 137 meters above sea level. In addition, I also own a genuine membership card with the number 137:
Finally, a problem
Let’s take a circle and divide its circumference into two parts according to the golden ratio. The golden ratio means that: (a+b)/a = a/ b
How many degrees will the angle corresponding to the arc be?
I’ll tell you: it’s almost exactly that, off by just about half a degree. They say this angle is characteristic of the arrangement of leaves in many plants (in Fibonacci phyllotaxis, the angle between successive leaves). And whether that’s humbug or just over-interpretation—does it really matter?
Recommended reading
M. Naschie, S. Olsen, J. He, S. Nada, L. Marek-Crnjac and A. Helal, “On the Need for Fractal Logic in High Energy Quantum Physics,” International Journal of Modern Nonlinear Theory and Application, Vol. 1 No. 3, 2012, pp. 84-92. doi: 10.4236/ijmnta.2012.13012.