The result of the distributed computing projectWieferich@Home
is presented: the binary periodic numbers of bit pseudo-length j ≤ 3500 obtained
by replication of a bit string of bit pseudo-length k ≤ 24 and increased
by one are Wieferich primes only for the cases of 1092 or 3510

Dobeš, Jan

Kureš, Miroslav

Bulgarian Digital Mathematics Library at IMI-BAS

Serdica J. Computing 4 (2010), 293–300SEARCH FOR WIEFERICH PRIMES THROUGH THE USEOF PERIODIC BINARY STRINGSJan Dobesˇ, Miroslav KuresˇAbstract. The result of the distributed computing projectWieferich@Homeis presented: the binary periodic numbers of bit pseudo-length j ≤ 3500 ob-tained by replication of a bit string of bit pseudo-length k ≤ 24 and increasedby one are Wieferich primes only for the cases of 1092 or 3510.1. Introduction. Let p be a Wieferich prime. A main motivationfor this research is the periodicity of the binary expression of p − 1 (for knownWieferich primes). The question of whether or not this phenomenon necessarilyfollows from properties of Wieferich primes is open. If we scan and test suchperiodic numbers, a need of knowledge about their quantity emerges, especiallyfor computer processing. We have answered the problem of a number of B-periodic numbers of given bit pseudo-length n in Proposition 1 and present twotables with some values.ACM Computing Classification System (1998): I.1.4, J.2.Key words: Wieferich prime, periodic bit string, distributed computing.294 Jan Dobesˇ, Miroslav KuresˇAfter a brief introduction to Wieferich primes, the main result is pre-sented. Within the distributed computing project Wieferich@Home, the authorshave proved (case-by-case, by computer) that the binary periodic numbers of bitpseudo-length j ≤ 3500 obtained by replication of a bit string of bit pseudo-length k ≤ 24 and increased by one are Wieferich primes only for the cases of1092 or 3510; i.e., there is no new Wieferich prime with the described propertyin the scanned area.2. Number of periodic strings. The numbers c(n) of aperiodicbinary strings of length n, n ∈ N, are tabulated in the OEIS database as thesequence A0275375, see [4]. Nevertheless, the explicit formula is also known, cf.e.g. [3]:c(n) =∑d|nµ(d)2nd ,where µ is the Mo¨bius function.Let us consider a binary string a of the forma = akak−1 . . . a1,a1, . . . , ak ∈ {0, 1}, in which a1 = 0 and at least one value is non-zero. Hereaftera number x ∈ N will be called a B-periodic number, if its binary representationx is a replication of such a string:a . . . a = x = xjxj−1 . . . x1,x1, . . . , xj ∈ {0, 1}. Left zeros can occur in x and j is a proper multiple of k,j > k. The number j will be called the bit pseudo-length of x (as the true bitlength can be lower just because of possible left zeros).Example 1. The smallest B-periodic number is 10, because its binaryrepresentation is10|10.Further examples of B-periodic numbers are e.g. 1092 with binary representation0100|0100|0100and 3510 with binary representation110|110|110|110.Search for Wieferich Primes through the Use of Periodic Binary Strings 295Let us denote by τ(n) the number of B-periodic numbers of bit pseudo-length n. It is evident that τ(n) = 0 if n is a prime (and, of course, if n = 1);hence the first non-zero value for τ(n) arises with n = 4. Further, let us denoteT (n) as a “cumulative τ(n)”, i.e.T (n) =n∑i=4τ(i).Proposition 1. The number of B-periodic numbers of bit pseudo-lengthn equalsτ(n) =122n −∑d|nµ(d)2nd − 2 .P r o o f. Very easy: the number of all periodic strings of bit pseudo-length n must be 2n − c(n). Then we exclude two periodic strings obtained byreplication of one digit strings: the string 111 . . . 1 (its last digit is not 0) and thestring 000 . . . 0 (it does not contain any 1). Exactly half of remaining periodicstrings have 0 as their last digit. Hence τ(n) =2n − c(n)− 22. So we can present the values for small n.n τ(n) T (n)4 1 15 0 16 4 57 0 58 7 129 3 1510 16 3111 0 3112 37 68And now, some higher (approximate) values:296 Jan Dobesˇ, Miroslav Kuresˇn τ(n) T (n)500 9.0463 · 1074 1.8093 · 10751000 1.6367 · 10150 3.2734 · 101501500 2.9612 · 10225 5.9224 · 102252000 5.3576 · 10300 1.0715 · 103012500 9.6931 · 10375 1.9386 · 103763000 1.7537 · 10451 3.5075 · 104513500 3.1729 · 10526 6.3459 · 105263. Wieferich primes. A Wieferich prime is a prime p which is asolution to the congruence equation2p−1 ≡ 1 (mod p2).ArthurWieferich has demonstrated a relation between the described primesand the most famous mathematical question (answered by Andrew Wiles on June23, 1993), Fermat’s Last Theorem.Theorem 1 (Wieferich 1909). Let p be an odd prime, and let a, b, c benon-zero integers such that ap + bp + cp = 0. Furthermore, assume that p doesnot divide the product abc. Then p is a Wieferich prime.There are only two known Wieferich primes, p1 = 1093 and p2 = 3511; aswe have seen, the numbers p1−1 and p2−1 are both B-periodic. This interestingproperty has been observed by Wells Johnson [2]. Furthermore, it is not known asyet if there exist infinitely or finitely many Wieferich primes (or even only thesetwo). So a conjecture that all Wieferich primes have this “periodic” property isreasoned and dramatic.Let us observe:Example 2. It is not true that a B-periodic number increment by oneis always a prime number. We have110|110 = 54as the fifth smallest B-periodic number and 54 + 1 is not a prime.Search for Wieferich Primes through the Use of Periodic Binary Strings 297Near-Wieferich primes are defined as primes having the form Z + Ap(mod p2) with both Z and A reduced modulo p values Z = ±1 and |A| ≤ 100.(128 instead of 100 appears in the Wieferich@Home application.)4. Search for Wieferich primes: the periodic test in Wiefe-rich@Home. The Wieferich prime 1093 was discovered by Waldemar Meissnerin 1913 and the Wieferich prime 3511 was discovered by Nicolaas G. W. H. Beegerin 1922. There are projects searching for new Wieferich primes. The newest resultis:Theorem 2 (Dorais and Klyve 2009, [1]). Up to 6.7 ·1015, only 1093 and3511 are Wieferich primes.The authors have produced their own original project Wieferich@Home,[5], as a distributed computing project publicly launched on December 29, 2007.The application contains two search algorithms: the complete test and the pe-riodic test. While the complete test tests linearly each successive prime, theperiodic test takes B-periodic numbers and checks if these numbers satisfy thecongruence condition. The algorithm for the periodic test works with numbers ofbit pseudo-lengths up to 3500. It follows that there are T (3500) ≈ 6.3459 · 10526B-periodic numbers to scan. The non-primality mentioned in Example 2 suggestsa sieving of these numbers.5. Quick Division Test (QDT). Testing B-periodic numbers with-out pre-test filters of non-prime numbers could slow down the computing anddecrease the count of tested B-periodic numbers. These cases can improve bythe following algorithm. The Quick Division Test (QDT) algorithm with a bitparser prepares numbers (j-bit pseudo-length of B-periodic numbers) for the con-gruence equation (CEq in algorithm). The arrayMn contains the first 9 Mersenneprimes. Finally, B-periodic numbers are increased by 1 and these potential primenumbers are tested by the congruence equation.Require: B-periodic numberEnsure: true, if input is not divisible by 3, 5, 7, 31, 127, ..., 2305843009213693951if B is not divisible by 5 thenfor i = 1 to 9 doe⇐ Mn[i]298 Jan Dobesˇ, Miroslav Kuresˇc⇐ n[i]s⇐ 1u⇐ 0d⇐ 0for j = 1 to the bit pseudo-length(B) dot⇐ bit(B, j)if s ≤ c thend⇐ d+ t · 2s−1s⇐ s+ 1elses⇐ 2u⇐ u+ dd⇐ tend ifend forif (u+ d) mod e ≡ 0 thenreturn falseend ifend forelsereturn falseend ifreturn true6. Periodic Test (PT). As to the pseudo-code of the algorithm, B-periodic numbers of bit pseudo-length 50 ≤ n ≤ 3500 are tested. B-periodicnumbers of bit pseudo-length n < 50 are ignored, because these primes weretested already by usual linear tests. The congruence equation (CEq) for our pur-poses returns attributes: 1 — Wieferich prime, 2 — near Wieferich prime. Forprocessing big integers of bit pseudo-length i > 64 we can use multiple-precisionarithmetic.Require: B — period, m — period count{big integer}{built-up B-period number from m-period B}p⇐ Bnumber(B,m)Search for Wieferich Primes through the Use of Periodic Binary Strings 299n⇐ bitlength(P )while n ≤ 3500 dop⇐ Bnumber(B,m){make odd number}p⇐ p+ 1n⇐ bitlength(p){numbers with pseudo-length n ≤ 50 were already tested}if 50 ≤ n ≤ 3500 then{Quick Division Test}if QDT(p) then{C: 1 — Wieferich prime, 2 — near Wieferich prime}C ⇐ CEq(p)end ifend if{next B-period}m⇐ m+ 1end whileWe repeat that numbers of a bit pseudo-length up to 50 were scannedby several authors, see e.g. [1] and references mentioned there. With the aidof the periodic test, the first findings for a publishing present an improvementsummarized in the following theorem.Theorem 3. Let n be a B-periodic number of a bit pseudo-length j ≤3500 obtained by a replication of a bit string of bit pseudo-length k ≤ 24. If n+1is a Wieferich prime, then n = 1092 or n = 3510.REFERENCES[1] Dorais F. G., D. W. Klyve. Near Wieferich Primes up to 6.7 · 1015.http://www-personal.umich.edu/~dorais/docs/wieferich.pdf[2] Johnson W. On the non-vanishing Fermat quotients. Journal fu¨r die reineund angewandte Mathematik, 292 (1977), 196–200.[3] Lothaire M. Combinatorics on words. Addison-Wesley, Reading MA, 1983.300 Jan Dobesˇ, Miroslav Kuresˇ[4] The On-Line Encyclopedia of Integer Sequences.http://www.research.att.com/~njas/sequences/[5] Wieferich@Home.http://www.elmath.orgJan DobesˇSoftware DevelopmentJira´skova 222/V38001 Dacˇice, Czech Republice-mail: info@dobesoft.czMiroslav KuresˇInstitute of MathematicsBrno University of TechnologyTechnicka´ 261669 Brno, Czech Republice-mail: kures@fme.vutbr.czReceived October 10, 2009Final Accepted March 11, 2010

Search for Wieferich Primes through the use of Periodic Binary Strings

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Search for Wieferich Primes through the use of Periodic Binary Strings

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