Russian Mathematician: Maurice Kraitchik and the New Mersenne Conjecture

Maurice Kraitchik (1882 - 1957) was a mathematician born in Russia who concentrated mainly on recreational aspects of the field. He did a great deal to popularize mathematics during his lifetime by writing many books on number theory and recreational math. He also worked as head editor for the math journal, Sphinx, from 1931 to 1939. Although Kraitchik was born in Russia, he grew up in Belgium and later moved to the United States. In the 1940s, he gave lectures on recreational math topics at the New School for Social Research in New York.

Today Kraitchik is probably most well-known for proposing a puzzle in his book La mathématique des jeux which inspired the "Two Envelopes Problem" - a paradoxical puzzle in the field of probability theory (which I don't have space enough here to fully explain).

Another problem (although much less well-known) that Kraitchik put forth was a conjecture in number theory involving Mersenne primes. (The French monk, Marin Mersenne, claimed in 1644 in his book Cogitata Physico-Mathematica that if p = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, and 257, then 2^p -1 is prime; and as a result, primes of the form 2^p-1 are now called 'Mersenne primes' - even though Marin got two of the previous values wrong and left out others. The current world-record prime is, 2^32582657-1, a Mersenne having 9,808,358 digits.)

Kraitchik's conjecture concerning Mersenne primes states that if 2ⁿ-1 is prime, then the number (2ⁿ+1)/3 will also be prime. But this is wrong. The first value that goes awry is n = 89.

A computer search reveals other counterexamples to be

107, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, ...

And then the sequence continues with the n values for the currently known Mersenne primes.

However, there is good news. After mathematicians refuted Kraitchik's conjecture, a few noticed a different property. Paul Selfridge, Stan Wagstaff, and Paul Bateman claimed that the following "new" Mersenne conjecture was true.

Let p be an odd number. If two of the statements below are true, then the third will be true as well.

1. 2^p-1 is prime (Mersenne)

2. (2^p +1 )/3 is prime (Kraitchik)

3. p has the form 2^k +/- 1, or 4^k +/- 3 (Selfridge, Wagstaff, Bateman)

The "+/-" symbol means that the number following them can be either positive or negative.

So far, this new Mersenne conjecture has been tested up to p = 13466917 and remains true.

Concerning Kraitchik's original conjecture, I have this new one that runs counter to it:

If p is an odd prime greater than 127, then 2^p-1 and (2^p+1)/3 will both never be prime for the same value of p. Or to say it another way, there will be no more primes p such that both 2^p-1 and (2^p+1)/3 are simultaneously prime for any p > 127.

If you are interested in acquiring a few of Kraitchik's books on number theory and recreational math, here is a brief list:

Mathematical recreations,

La mathématique des jeux ou Récréations mathématiques,

Recherches sur la théorie des nombres

Théorie des Nombres

. Paris: Gauthier-Villars, 1922. . Paris: Gauthier-Villars, 1924 Paris: Vuibert, 1930, 566 pages. London: George Allen & Unwill Ltd, 1955 and New York: Dover, 1953.