Concerning the Numbers 22p + 1, p Prime

Transcription

1 Conerning the Numbers 22p + 1, p Prime By John Brillhart 1. Introdution. In a reent investigation [7] the problem of fatoring numbers of the form 22p + 1, p a, was enountered. Sine 22p + 1 = (2P - 2*<p+1) +1) (2P + 2è(p+1) + 1 ) for odd p, the problem onsists of fatoring the two trinomials on the right. In this paper the results of a searh for fators of these trinomials are given, as well as a determination of the nature of ertain of these numbers for whih no fator was found. 2. Elementary fators. Let Np = (2" - 2i(p+I> + 1) (2" + 2è(p+1) + 1) = Ap Bp, p an odd. A. From the fat that 5 Np, it easily follows that 5 Ap iff p = ±1 (mod 8) and 5 Bp iff p = ±3 (mod 8). On the other hand, 52 \ Np unless p = 5; for, sine 2 is a primitive root of 25, 2 belongs to the exponent <j> (25) = 20. But 22p = 1 (mod 25), or 24p m 1 (mod 25). Therefore, 20 4p, or p = 5. Thus, if p = 5, = 1025, while if p ^ 5, 52 \ Np. B. If g is a 5^5 and q \ Np, then 24p = 1 (mod q). But then 2 belongs to the exponent 4p (mod q). Thus by Fermat's Theorem, 4p q 1 ; that is, every divisor 5^5 of Ap or Bp is =1 (mod 4p). C. Suppose p is odd and q = 4p + 1 is a. Then 2q~i = 24p = 1 (mod q). It follows from Euler's Criterion that 22p = (-) (mod q). But sine p is odd, q = 5 (mod 8). Therefore, 22p = -1 (mod q), or q 22p + 1. Unfortunately, however, it has not been possible to disover the onditions that determine whih of Ap and Bp q will divide. 3. The Searh. A. Extent. The searh for fators q 5 of AP and Bp, whih was onduted on the IBM 701 at the University of California, Berkeley, was made over the following intervals: 1 < q < y/tfa for Bm 1 < q < for An 1 < q < 230 for 71 < p ^ 179 and p = < q < 228 for 179 < p < 1200, p * 241. No Np for p < 71, p t^ 59, were onsidered, sine these numbers have been ompletely fatored. A/24i was examined along with AV3 to the bound 230, these numbers being of partiular interest (See [7]). B. Results, (i) The program produed a vast number of new fators, as well as several orretions to the literature (See [4]). The new fators of Np, p < 250, are indiated in the aompanying table by * to distinguish them from fators pre- Reeived January 10, Liense or opyright restritions may apply to redistribution; see

2 CONCERNING THE NUMBERS 22p + 1, p PRIME 425 viously known [2]. For 250 < p < 1200 all fators > 300,000 are new, and are therefore not indiated by. A dot following the final fator means that the nature of the omplementary fator is unknown. (ii) A omplete fatorization was aomplished for BM, AS3, and Am, the primality of the omplementary fator in eah ase being assured by the non-existene of a fator below its square root. The fatorization of BM is of partiular interest, sine this number appears in [2] and [3] as a. The author would like to thank Mr. K. R. Isemonger for providing the omplete fatorization of B97, as well as the muh sought after fatorization for A 71, whih, previous to his attak on the number, had only been known to fator into the produt of two s. (iii) A program was written to test the divisibility and multipliity of all known fators, with the result that all fators were found to be orret, but none was found to be multiple. C. The Program. The struture of the searh program was similar to that desribed in [1]. In partiular, for eah p a table of differenes was omputed from the first 1155 = terms of the sequene 4pf + 1, k = 1, 2,, that remained after the multiples of 3, 5, 7, and 11 had been sieved out. This table was used repeatedly by the program to produe a sequene of trial divisors, among whih the fators, if any, were to be found. The remainders of Ap and Bp for eah trial divisor were alulated by residue methods, both remainders being alulated at the same time beause of the similarity in form of Ap and Bp. The ourrene of a 0 remainder in this alulation signalled the disovery of a fator of one of the two numbers, but not both, sine obviously they are relatively. To examine eah Np required from 5 to 15 minutes, the N for the larger p's requiring a shorter time. 4. Primality Testing. A. At the onlusion of the searh for fators, the primality of several numbers of immediate interest, namely, An and Am, was still in doubt, beause no fator had been found. It was then noted by Professor D. H. Lehmer that the primality of numbers of the form under onsideration ould be deided by Proth's Theorem [5]: "If M = fe-2* + 1, where 0 < k < 2", and (~J = - 1, then M is iff akm-i> m _ l (mod M) In the present ase AP,BP = M = (2Up~1) ± 1 ) 2*(p+1) + 1, with 0 < k = 2S(P_1) ± 1 < 2i(p+1) for p an odd, the value of a being easily obtained from the reiproity law for the Jaobi symbol. A program was aordingly written by Professor Lehmer for the IBM 701 to alulate the required residues. The modulus used for eah test was Np rather than the Ap or Sp in question, so that the redution of the suessive powers ould be aomplished by multi-preision subtration instead of division by a multi-preision divisor. The remainder thus produed was further redued mod Ap or Bp by a subtrative routine written by the author. The final residues in binary from both routines have been preserved on IBM ards for later heking purposes. B. It is believed that the two testing programs were aurate, sine the antiipated results were obtained in every trial ase save one. In this ase, B59, a disrepany existed between the literature, whih stated the number was, and the Liense or opyright restritions may apply to redistribution; see

3 426 JOHN BRILLHART Table of Fators 2p _ 2*<p+» + 1 2p + 2*<p+1> * * * * * * * * * * * * * * * * * * * * * *- Liense or opyright restritions may apply to redistribution; see

4 CONCERNING THE NUMBERS 22p + 1, p PRIME 427 Table of Fators Continued 2" 2Kp+i) - _ i 2? -f- 2Hp+1) * * * * Liense or opyright restritions may apply to redistribution; see

5 428 JOHN BRILLHART Table of Fators Continued 2p 2*<*+1> + 1 2p -f 2*<p+» Liense or opyright restritions may apply to redistribution; see

6 CONCERNING THE NUMBERS 22p + 1, p PRIME Table of Fators Continued V i> 2*<p+1> É p -f- 2}("+1) Liense or opyright restritions may apply to redistribution; see

7 430 JOHN BRILLIERT test routine, whih stated the opposite. The number was immediately run on the fatoring program, and muh to the satisfation of all onerned, a fator was found, and the test routine was exonerated. A further verifiation of a kind has ome from Mr. Isemonger, who, ating on the test results that An and B97 were omposite, sueeded in finding the fatorizations mentioned above. C. All Ap and Bp, 71 ^ p ^ 757, for whih no elementary or other fator was known, were tested for primality. In all, 50 numbers were tested, with the result that 14 of them were found to be. These are listed as in the aompanying table, while the remaining 36 omposite numbers are indiated as suh by a "" in the proper positions of the table. Eah number with 71 : p ^ 457 was tested twie with omplete agreement in the results. No number for p > 457 was tested twie, for testing a single number in this range required approximately 30 minutes. 5. Aknowledgements. The author would like to express his gratitude to Professor Lehmer for his very generous ontributions of time and effort in onstruting the primality test, whih has brought this paper to suh a satisfatory onlusion. In addition, he would like to thank Dr. John Selfridge for his areful reading of the preliminary manusript, and Mr. Vane Vaughan and Robert Innes for their assistane in the prodution phase of the program. University of San Franiso San Franiso, California 1. John Brillhart & G. D. Johnson, "On the fators of ertain Mersenne numbers," Math. Comp., v. 14, 1960, p. 2. A. J. C. Cunningham & H. J. Woodall, Fatorizations of (yn =F 1), Hodgson, London, 1925, p M. Kraithik, Reherhes sur la Théorie des Nombres, Tome II, Paris, Counil Bulletin, Washington, 1941, p , F. Proth, "Théorèmes sur les nombres premiers," C. R. Aad. Sei. Paris, v. 87, 1878, 4. D. H. Lehmer, Guide to the Tables in the Theory of Numbers, National Researh p R. M. Robinson, "Some fatorizations of numbers of the form 2" ± 1," MTAC, v. 11, 1957, p Robert Spira, "The omplex sum of divisors," Amer. Math. Monthly, v. 68, 1961, p Liense or opyright restritions may apply to redistribution; see