MERSENNE AND FERMAT NUMBERS

RAPHAEL M. ROBINSON

1. Mersenne numbers. The Mersenne numbers are of the form 2n-1. As a result of the computation described below, it can now be stated that the first seventeen primes of this form correspond to the following values of n:

2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281.

The first seventeen even perfect numbers are therefore obtained by substituting these values of n in the expression 2n-1(2n-1). The first twelve of the Mersenne primes have been known since 1914; the twelfth, 2127-1, was indeed found by Lucas as early as 1876, and for the next seventy-five years was the largest known prime. More details on the history of the Mersenne numbers may be found in Archibald ; see also Kraitchik . The next five Mersenne primes were found in 1952; they are at present the five largest known primes of any form. They were announced in Lehmer  and discussed by Uhler .
It is clear that 2n-1 can be factored algebraically if n is composite; hence 2n-1 cannot be prime unless n is prime. Fermat's theorem yields a factor of 2n-1 only when n+1 is prime, and hence does not determine any additional cases in which 2n-1 is known to be composite. On the other hand, it follows from Euler's criterion that if n == 0, 3 (mod 4) and 2n+1 is prime, then 2n+1 is a factor of 2n-1. Thus, in addition to cases in which n is composite, we see that 2n-1 is composite when 2n+1 is prime as well as n, provided that n == 3 (mod 4) and n > 3. Aside from this, factors of 2n-1 are known only in individual cases. If no factor is known, the best way to find out whether 2n-1 is prime is to apply a test due essentially to Lucas, but stated in a simplified form by Lehmer [6, Theorem 5.4].

THEOREM. Let S1 = 4, Sk+1 = Sk2-2. Then, for n > 2, the number 2n-1 is prime if and only if Sn-1 == 0 (mod 2n-1).

Alternatively, we may start with S1 = 10; or, if n == 3 (mod 4), we may also use S1 = 3. Such a test was first applied by Lucas in 1876 to show that 2127-1 is prime. By 1947, all of the numbers 2n-1 with n <= 257 had been tested; if there had been no errors in the computations, this would have completed the proof or disproof of the various cases of Mersenne's conjecture of 1644. In 1951, the first application of an electronic computer to testing Mersenne numbers for primeness was made by A. M. Turing at the University of Manchester; however, no new primes were found, and no remainders were saved for purposes of comparison.
In 1952, a program for testing Mersenne numbers for primeness on the SWAC (the National Bureau of Standards' Western Automatic Computer, at the Institute for Numerical Analysis in Los Angeles), planned and coded by the author, using Lucas's test, was carried out, with the cooperation of D. H. Lehmer and the staff of the I. N. A. My thanks are due especially to Emma Lehmer, who did various auxiliary computations, including checking some of the results obtained against earlier results. The program was first tried on the SWAC on January 30, and two new primes were found that day; three other primes were found on June 25, October 7, and October 9.
At that time, the total memory of the SWAC consisted of 256 words of 36 binary digits each, exclusive of the sign. For the Mersenne test, half of this memory was reserved for commands. Since successive squarings of numbers less than the modulus 2n-1 are required, this modulus was restricted to 64 words, so that the condition n < 64 * 36 = 2304 was imposed. The estimated running time for the program was 0.25n3+125n2 microseconds, and the actual time was in fair agreement with this. Thus, roughly speaking, the testing time was a minute for the first and an hour for the last of the five new primes. Each minute of machine time is equivalent to more than a year's work for a person using a desk calculator.
The output of the SWAC for each n was the least non-negative residue of Sn-1 (mod 2n-1), written to the base 16. This was a long string of zeros if 2n-1 was prime, and otherwise was an apparently random sequence of digits. Every value of n < 2304, for which no factor of 2n-1 was known, was run twice on the SWAC; in case of disagreement, a third run was made. No remainder was accepted as correct until it had been obtained twice, and indeed on different days. (Although the bulk of this work was done in 1952, the checking was not complete in all respects until late in 1953. The submission of this report was delayed for this reason.) It is of course out of the question to print the hundreds of remainders obtained, but they will remain on file at the Institute for Numerical Analysis. [Added in proof. On July 1, 1954, the I.N.A. became a part of the University of California at Los Angeles; it is now known as Numerical Analysis Research.]
We shall confine ourselves here to mentioning various results of earlier computations which have been verified or contradicted. (In a few cases, earlier computers had used the test with S1 = 3; in these cases, the same computation was also made on the SWAC.) The six results of Uhler , for n=157, 167, 193, 199, 227, 229, were all checked by him by converting the corresponding SWAC remainders to decimal form, and found to be correct. The rest of the checking was done by Emma Lehmer. The three results of D. H. Lehmer , for n = 139, 149, 257, were all verified. (Although Lehmer's remainder for n = 139 had not been published, it was available for checking.) The result of Powers  for n=241 was found to be correct; the remainders from his earlier computations, for n = 103, 109, had not been published, and were not available for checking. On the other hand, the four remainders of Fauquembergue , for n=101, 103, 109, 137 were all found to be incorrect, as was the remainder of Barker  for n = 167.
If 2n-1 is prime, so that Sn-1 == 0 (mod 2n-1), then Sn-2 is a solution of the congruence x2 == 2 (mod 2n-1). Thus Sn-2 == +- 2(n+1)/2 (mod 2n-1). The calculation of the next to the last remainder Sn-2 was undertaken for the various known Mersenne primes, both with S1 = 4 and with S1 = 10, with the following results for the sign in the above congruence:

```        n=3,  5,  7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607,1279,2203,2281
S=4    +   +   -   +   -   -   +   +   -   -    +    -    -    -    +    -
S=1O   -   -   -   +   +   +   +   +   +   +    +    +    -    -    -    +
```

We have carried the program of testing Mersenne numbers for primeness about as far as is practicable using present day computers. The smallest untested Mersenne number is 22309-1, which does not appear to be a case of exceptional interest. A more interesting case is 28191-1, which should be prime according to a conjecture that 2n-1 is always prime when n is a Mersenne prime. This conjecture is true for the first four cases, corresponding to the exponents

3 = 22-1, 7 = 23-1, 31 = 25-1, and 127 = 27-1.

The fifth case corresponds to n = 8191 = 213-1. The corresponding Mersenne number was actually tested in 1953 by D.J. Wheeler on the Illiac, at the University of Illinois, one hundred hours of machine time being required. The remainder obtained was not zero, indicating that the number is composite, and the conjecture therefore false. According to Dr. Wheeler, considerable confidence may be placed in this result, since the computation was carefully checked.

2. Fermat numbers. The Fermat numbers are of the form 2n+1. The only new result obtained here was that 21024+1 is composite, and hence that a regular polygon with this number of sides cannot be constructed with ruler and compass.
The number 2n+1 can be factored algebraically unless n is a power of 2. Factors are known in a few other cases. If no factor is known, the following test may be used.

THEOREM. If n > 1, then 2n+1 is prime if and only if 32n-1 == -1 (mod 2n+1).

The program set up for testing Mersenne numbers on the SWAC was modified to apply to Fermat numbers. The range for the exponent n was the same, but with n = 2m, this yields m <= 11. Now 22m+1 is prime for m = 0, 1, 2, 3, 4, and factors were known for m = 5, 6, 9, 11. The Fermat numbers corresponding to m = 7, 8 had been proved composite by Morehead and Western [8;9], and the remainders which they gave were found to be correct. (The necessary conversion of the SWAC result to decimal form was done by Emma Lehmer.) In the one new case, m = 10, the least positive residue of 321023 (mod 21024+1) was found to be

```                   8x   4z258xu89   uw71y6w35   9z1vyy4u5   498v2v7v7
55y9wy98v   6yx3yy0x4   00u07877w   2866316zu   85wy92558   3y201x7xO
04w1zv076   yu292wxx4   0502v7567   226047037   308u12z32   887vyxu4x
51w50169z   6815w5Ou1   0wy653448   v953xy0w6   8uv4492z2   v1u564ux9
494x25wv7   wux26uuvw   4w5x12730   622y6z435   5xy035xx2   8798y8098
```

to the base 16, using 0,1,2,3,4,5,6,7,8,9,u,v,w,x,y,z as digits. Thus 2 +1 is composite. This result was first obtained in February 1952. As an extra check in this case, the test was recoded by Emma Lehmer; the modified test was run in January 1953, and the above remainder was verified.
Later in 1953, Selfridge  showed that this Fermat number has the factor 11131*212+1, which confirms the above result, but in a sense renders it obsolete even before it is submitted for publication. Selfridge also found a factor of 2216+1. Previously known factors of Fermat numbers may be found in Kraitchik . Factors of 22m+1 are now known for m = 5, 6, 9, 10, 11, 12, 15, 16, 18, 23, 36, 38, 73.
The first Fermat number of unknown character is 28192+1, corresponding to m=13. The difficulty of testing this number is about the same as for the Mersenne number 28191-1. It would probably be considerably easier to find some additional factors of Fermat numbers by trial.

REFERENCES

1. R. C. Archibald, Mersenne's numbers, Scripta Mathematica vol. 3 (1935) pp. 112-119.
2. C. B. Barker, Proof that the Mersenne number M167 is composite, Bull. Amer. Math. Soc. vol. 51 (1945) p. 389.
3. E. Fauquembergue, Nombres de Mersenne, Sphinx-OEdipe, vol. 9 (1914) pp. 103-105; vol. 15 (1920) pp. 17-18.
4. M. Kraitchik, On the factorization of 2n+-1, Scripta Mathematica vol. 18 (1952) pp. 39-52.
5. D. H. Lehmer, Note on the Mersenne number 2139-1, Bull. Amer. Math. Soc. vol. 32 (1926) p. 552; Note on Mersenne numbers, ibid. vol. 38 (1932) p. 383-384.
6. ------, An extended theory of Lucas' functions, Ann. of Math. vol. 31 (1930) pp. 419-448.
7. ------, Recent discoveries of large primes, Mathematical Tables and Other Aids to Computation vol. 6 (1952) p. 61; A new Mersenne prime, ibid. p. 205; Two new Mersenne primes, ibid. vol. 7 (1953) p. 72.
8. J. C. Morehead, Note on Fermat's numbers, Bull. Amer. Math. Soc. vol. 11 (1905) pp. 543-545.
9. J. C. Morehead and A. E. Western, Note on Fermat's numbers, Bull. Amer. Math. Soc. vol. 16 (1909) pp. 1-6.
10. R. E. Powers, Certain composite Mersenne's numbers, Proc. London Math. Soc. (2) vol. 15 (1916) p. xxii; Note on a Mersenne number, Bull. Amer. Math. Soc. vol. 40 (1934) p. 883.
11. J. L. Selfridge, Factors of Fermat numbers, Mathematical Tables and Other Aids to Computation vol. 7 (1953) pp. 274-275.
12. H. S. Uhler, First proof that the Mersenne number M157 is composite, Proc. Nat. Acad. Sci. U.S..A. vol. 30 (1944) pp. 314-316; On all of Mersenne's numbers particularly M193, ibid. vol. 34 (1948) pp. 102-103; Note on the Mersenne numbers M157 and M167, Bull. Amer. Math. Soc. vol. 52 (1946) p. 178; On Mersenne's number M199 and Lucas's sequences, ibid. vol. 53 (1947) pp. 163-164; On Mersenne's number M227 and cognate data, ibid. vol. 54 (1948) p. 379; A new result concerning a Mersenne number, Mathematical Tables and Other Aids to Computation vol. 2 (1946) p. 94.
13. ------, A brief history of the investigation on Mersenne's numbers and the latest immense primes, Scripta Mathematica vol. 18 (1952) pp. 122-131; On the 16th and 17th perfect numbers, ibid. vol. 19 (1953) pp. 128-131.

UNIVERSITY OF CALIFORNIA, BERKELEY

Received by the editors February 7, 1954.