BY HORACE S. UHLER

THIS ARTICLE IS ADDRESSED TO THOSE VALUED READERS WHO PREFER OR REQUIRE A SYNOPSIS OF FACTS BEFORE ALGEBRAIC PROOFS.

What are Mersenne numbers and why should any television fan abandon this latest relaxation fad for neglecting the proper maintenance of muscular tissue in order to consume any mental energy on something as ancient, trite, and obvious as any kind of number? We shall endeavor to give a succinct answer to the first part of this question while abandoning the second part to the anthropologist, to the student of the completely unreliable mammal often questionably referred to as "homo sapiens."

It is convenient to call any number expressed by the form

When[1] did Marin Mersenne live? He was born near Oizé (Sarthe) on Sept. 8, 1588, and died in Paris on Sept. 1, 1648. Mersenne and Descartes were fellow students at the Jesuit college of La Flèche. In 1611 Mersenne joined the Minim Friars, and in 1620 he made permanent residence in Paris at the convent of L'Annonciade. The Minimi[2] (or Minims) were members of a Roman Catholic monastic order founded in Italy in 1435 by St. Francis of Paula. This designation was assumed as an expression of self-abasement. A portrait of Father Mersenne is reproduced on page 72 of Oystein Ore's fascinating text entitled

What interest attaches to integers of the form 2^

In this connection the following quotations are interesting and informative. From pages 43 and 44 of Peter Barlow's book entitled

He says (in German): "This ancient idea of the perfect number and its associated questions are not especially important, I treat these matters only because by so doing we shall immediately encounter two problems which are unsolved even to-day: ... [1927]. Is the number of even perfect numbers infite? I do not know. Is the number of odd perfect numbers infite? I don't even know whether a single odd perfect number exists. Nevertheless I beg the reader not to ponder long over these two questions; he or she will learn about many more abstruse and promising problems during the study of this work."

In marked contrast with the last opinion we read from the pen of E. T. Bell:[4] "This raises the interesting and extremely difficult question of finding those primes

Greater human interest in Mersenne numbers developed from his apparently off-hand prediction that the only values of

In critical vein attention may be called to a Special to the

An extension of the playground belonging to the perfect numbers is afforded by the multiply perfect numbers which are defined as having the sum of their divisors equal to a multiple (2, 3, 4, . . . ) of the number itself. For example 672 = 2^5*3*7 is a number of this kind since the sum of its 192 divisors equals 2*672. The multiple may be called the

The reader who desires full details concerning the names of the investigators of the prime or composite character of Mersenne numbers, of the factors when obtainable, of the dates and journals of publication, etc., should refer to the characteristically complete and reliable paper by R. C. Archibald in SCRIPTA MATHEMATICA, v. 3, No. 2, p. 112-119, April 1935. Since Archibald's paper practically exhausts the references up to the last date we shall continue the present record after the year 1935.

At that time there were six values of

Why is it so difficult to test with desk machines a given number of the form

An extremely simple numerical illustration should suffice. Consider the case where

Too much emphasis cannot be placed upon the fact that at long last the courageous arithmetician has been emancipated forever from the slavery of hand computation on Mersenne and analogous numbers by the appearance on Earth of the magnificent digital calculating assemblages. And in point of time these are still in their infancy. As stated in an earlier paragraph the SWAC battery proved on the night of January 30, 1952, that 2^521 - 1 and 2^607 - 1 are Mersenne primes. Behold

Finally and for completeness I squared 2^606 and multiplied by 2 to get 2^1213 with the object of computing the largest perfect number known to man at this time.

The SWAC digital computer uses the base 16 instead of the radix 10 as may be inferred from the fact that the remainders kindly sent to the writer by D. H. Lehmer were expressed in terms of 2^4. For example the SWAC gave for the last remainder for

The probability that a computer would obtain the (

The only reason for relegating this paragraph to the end of the historiette is that it refers chiefly to new prime numbers which are not of the Mersenne type. From various sources,[10] including personal letters from England, the following valuable information has been gleaned. A. Ferrier proved that (2^148 + 1)/17 is prime. When written out on the base ten we find the 44-digit number 2098 89366 57440 58648 61512 64256 61022 25938 63921. Using the EDSAC, Miller and Wheeler have demonstrated that the expression

After the preceding manuscript had been received by the Editor-in-Chief of SCRIPTA MATHEMATICA I became aware of the fact that valuable information acquired subsequently from various sources concerning the work with SWAC should be included in this paper provided that no unprofessional duplication be thereby incurred. Hence I wrote to headquarters, that is to my friend Professor D. H. Lehmer, Director of Research of the Institute for Numerical Analysis, National Bureau of Standards, Los Angeles, California. His answering letter was immediate and it contained without reservation the following up-to-date minute details, as of June 4, 1952.

All Lucas tests that were performed earlier with hand machines have been repeated on the SWAC. The machine definitively emphasized Barker's fiasco. A few other disagreements will be investigated by Lehmer.

"The SWAC Mersenne Project consists in testing 2^

"The 'credit' for these results goes to the SWAC." Personal credit belongs to Professor R. M. Robinson of the University of California, at Berkeley, who designed this program.

As to prime factoring the SWAC has recently proved that "2^89 + 1 = 3 * 179 * 62020897 * 18584774046020617," a new valuable result.

In an earlier paper[11] we have recorded the numbers of digits in

My indebtedness to Professor Lehmer is appreciably enhanced and gratefully acknowledged because of the following additional information which he imparted to me in a letter dated June 16, 1952.

"The reason why

Paging Peter Barlow, Esq.! On August 21, 1952, the present author received from Professor D. H. Lehmer a note containing the following very exciting important facts: On June 25 the SWAC discovered that the Mersenne-like number 2^1279 - 1 is prime. "We have tested this number quite a number of times since. At high speed it takes less than 13 1/2 minutes. You may be interested in knowing that the 1277th term of the sequence 4, 14, ... is congruent to -2^640 mod (2^1279 - 1)." The last item may be looked upon as an additional check but the minus sign alone is informative.

The writer immediately began to calculate on the base ten the exact values of

The first calculation of

The accuracy of this value of

YALE UNIVERSITY

1 Encyc. Brit., 14th ed., v. 15, p. 284.

2 Ibid., p. 541.

3 Edmund Landau, Elementare Zahlentheorie, Chelsea Publishing Co. (1946), p. 18, 19.

4 E. T. Bell, Numerology (1933), p. 185, 186.

5 H. S. Uhler, "On Mersenne's Number M227 and Cognate Data, Bull. Am. Math. Soc., v. 54, no. 4 (April 1948) p. 378-380.

6 E. Fauquembergue, Sphinx OEdipe, v. 9, (June 1914) p. 85 103-105.

7 D. H. Lehmer, "An Extended Theory of Lucas' Functions," Thesis at Brown University, (Dec. 26, 1929) p. 443-445.

8 H. S. Uhler, "The Magnitude of Higher Terms of the Lucasian Sequence 4, 14, 194, ...," Math. Tables and other Aids to Computation, v. 3, no. 22 (April 1948), p. 142, 143.

9 C. B. Barker, "Proof That the Mersenne Number M167 Is Composite," Bull. Am. Math. Soc., v. 51 (1945) p. 389.

10 J. C. P. Miller and D. J. Wheeler, "Large Prime Numbers," Nature, v. 168, (1951) p. 838.

J. C. P. Miller, "Large Primes," Eureka, no. 14 (1951) p. 10, 11.

D. H. Lehmer as Editor and Reviewer, M. T. A. C., v. 6, no. 37 (January 1952), p. 61.

11 H. S. Uhler, " Many-Figure Values of the Logarithms of the Year of Destiny and Other Constants," SCRIPTA MATHEMATICA, v. 17, nos. 3-4 (Sept.-Dec. 1951) p. 202-208.

Return to this paper's entry in Luke's Mersenne Library and Bibliography