## The Feynman Point & Other Curiosities

Math TriviaThe decimal expansion of p begins 3.14159265358 … and the number is defined as the ratio of a circle’s circumference to its diameter. If you fast forward to about the 760th digit in the decimal expansion of p, you will see the string “…34 999999 83…” That group of six repeating 9s in the center is known as the ‘Feynman Point.’

p is a real number that is irrational and transcendental, meaning it is not the ratio of two integers and it is not the solution of a polynomial with rational coefficients. Although it hasn’t been proved, it’s generally believed that any finite set of digits will eventually occur in the decimal expansion of any irrational number due to their non-repeating behavior. For example, it’s thought that at some point in the decimal expansion of, say, v666 that the string ‘6660066600666’ will occur, and the string ‘12345432123454321’ will occur, and any other finite string you come up with. Thus, the appearance of six 9s in the decimal expansion of p isn’t that unusual. But the fact that it occurs after only 762 digits is what makes it a genuine curiosity. (For comparison, the earliest location of any four repeating digits is at position 1589, in which four 7s appear.)

The group of six 9s in p referred to as the Feynman Point is named after Nobel Prize winning physicist Richard Feynman (1918 – 1988), who was awarded the prize in 1965 for his work on quantum electrodynamics. Feynman once stated in a lecture that he wished he could memorize all the digits of p up to the group of six 9s so he could continue: “nine, nine, nine, nine, nine, nine, and so on.” Of course that would be misleading because if the digits went on as repeating 9s, p would cease being an irrational number.

Quite a bit of research has been done on patterns that occur in the decimal expansions of irrational numbers. But if one becomes too intrigued with the digits for their own sake, it is not looked upon favorably by most mathematicians. In Petr Beckmann’s book, A History of p, he has a chapter titled “Digit Hunters,” and uses the phrase disparagingly since there is no practical use for computing more and more digits of p, or for locating patterns in their decimal expansions. (Only 10 digits are needed for any real-world problem).

Nevertheless, when you have a computer it’s fun to search for patterns in anything. So taking inspiration from the Feynman Point I thought of this sequence: The first position in the digits of p (not counting the initial 3) at which a string of n copies of the digit n occurs. For example, the first occurance of four 4s is at position 54525, “…11793 4444 82014…” and that is the fourth term of our sequence. Here are all the known terms: 1, 135, 1698, 54525, 24466, 252499, 3346228, 46663520, 564665206. Because all finite patterns of digits are believed to eventually appear in irrational numbers, surely this sequence is infinite. But where will the string of one hundred 100s occur?

I’ll close this article with two strange mathematical curiosities only loosely related to the Feynman Point.

Consider v666 = 25.8069758011278803151884206 … The first ‘666’ string occurs in its decimal expansion at position 503, and the locations for the next 666s are 928, 975, 2705, 4229, 7021, 9195, 9338, and 9349. Those are all positions of 666s among the first 10,000 digits, and I have dubbed it ‘Lucifer’s sequence’ because it is believed that if a person recites the terms backwards while spinning counterclockwise six times, upon their final revolution they will either become totally omniscient, or their head will burst into flames. Only kidding.

The second curiosity is based on primes, my favorite type of number, and we’ll use some information from the Feynman Point. The six repeating 9s start at the 762nd place and end at the 767th place. If we concatenate those three numbers we get this prime: 762999999767. And if we continuously add zeros between those three numbers we’ll continue to find more. The next one is 76200009999990000767. Let f(n) = (762 * 10n+6 + 999999) * 10n+3 + 767. It is prime for n = 0, 4, 11, 15, 32, 215, 408, 461, 1489, 9349, 9470 and no more up to 10000. The last two have 18710 and 18952 digits, respectively (but they’re actually only probable primes since their form is not easily provable). Surely there must be infinitely many Feynman Point primes.