DOCSLIB.ORG

RFSC 04-02 Revised a METHOD for GENERATING MERSENNE PRIMES and the EXTENT of the SEQUENCE of EVEN PERFECT NUMBERS Simon Davis Re

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

RFSC 04-02 Revised A METHOD FOR GENERATING MERSENNE PRIMES AND THE EXTENT OF THE SEQUENCE OF EVEN PERFECT NUMBERS Simon Davis Research Foundation of Southern California 8837 Villa La Jolla Drive #13595 La Jolla, CA 92039 Abstract. A condition is obtained for the generation of new Mersenne primes from a com- bination of Mersenne numbers with prime indices. It is veri¯ed that all known Mersenne p1 p2 prime indices greater than 19 have the form p1 + p2 ¡ 1, where 2 ¡ 1 is prime and 2 ¡ 1 is composite, and that all Mersenne prime indices less than or equal to 19 have the form 0 0 0 0 p1 p2 p1 + p2 ¡ 1, with 2 ¡ 1 and 2 ¡ 1 both being primes. Arithmetical sequences for the exponents of composite Mersenne numbers are obtained from partitions into consecutive integers, and congruence relations for products of two Mersenne numbers suggest the exis- tence of in¯nitely many composite integers of the form 2p ¡1 with p prime. The congruence properties of di®erences of powers with polynomial exponents are used to prove the in¯nite extent of the sequence of Mersenne primes. MSC: 11N13, 11N32, 11P83 1. Introduction The Lucas-Lehmer test [11][12], together with several characteristics of prime divisors of Mersenne numbers, which are valid for all Lucas sequences can be used to determine theoretically whether the integer 2n ¡1 is prime. The congruence modulo 2n ¡1 is satis¯ed by the known Mersenne primes, although the di±culty of the computation increases for larger values of n. The conjecture of the existence of in¯nitely many Mersenne primes and the problem of establishing the in¯nite extent of the sequence of composite Mersenne numbers with prime indices may be solved without the consideration of speci¯c values of the exponent. The generality of these statements allows for the proofs to be based on conditions imposed on integers of the same order as the exponents. It is shown in x2that since every prime exponent p has the form p1 + p2 ¡ 1, where p1 p1+p2¡1 and p2 are prime, with several conditions being given for 2 ¡ 1 to be composite, all known Mersenne primes greater than 219 ¡ 1 are shown to have exponents of the form p1 +p2 ¡1 where p1 is a Mersenne prime index and p2 is the index of a composite Mersenne number. The Mersenne numbers have a geometrical representation which may be used to derive congruence relations for compositeness based on the partition of the array representing 2n ¡ 1. The solutions to the congruence relations yield arithmetical sequences for the exponent. However, the greatest common denominator of the initial term and the di®erence can be equated with ord2m¡k(k) for some m, k, so that none of the integers in the sequence are prime. Nevertheless, this allows a characterization of the set of exponents greater than 6 of composite Mersenne numbers. The congruence relations for 2p1+p2¡1 ¡ 1 provide a further indication of the existence of in¯nitely many composite Mersenne numbers with prime exponents. A proof of the existence of in¯nitely many Mersenne primes is given in x4. The existence of a ¯nite number of prime solutions to af(n) ¡ bf(n) ´ 0 (mod n) when f(x) does not have a zero at x = 1, is used to demonstrate that there are an in¯nite number of values of n for which 2n ¡ 1 does not have a proper prime divisor. 2. The Exponents of Mersenne Numbers and Arithemetical Progressions There are two in¯nite sequences, of Mersenne numbers with odd index, and primes, in the arithmetical progression 6n + 1, n 2 Z, and the coincidences of these two sequences determine whether the set of even perfect numbers continues inde¯nitely. 1 Since 6n + 1 can be factorized only if n has the form 6xy § (x + y) with x and y, the Mersenne number 2p ¡ 1 is prime only if it equals 6n + 1; n = 6xy § (x + y) + z, z 6= 0 with 6xy § (x + y) + z 6= 6x0y0 § (x0 + y0) for any integers x0; y0. Given the condition p 2 ¡ (6z + 1) = (6x § 1)(6y § 1), consider two primes p1 and p2 such that p1 2 ¡ (6z1 + 1) = (6x1 § 1)(6y1 § 1) = 6h1 + 1 h1 = 6x1y1 § (x1 + y1) (2:1) p2 2 ¡ (6z2 + 1) = (6x2 § 1)(6y2 § 1) = 6h2 + 1 h2 = 6x2y2 § (x2 + y2) Multiplying these two integers gives p1+p2 2 ¡(6z1 +1)(6z2 +1)¡(6z1 +1)(6h2 +1)¡(6z2 +1)(6h1 +1) = (6h1 +1)(6h2 +1) (2:2) or equivalently p1+p2¡1 2 = [3(h1 + z1) + 1][6(h2 + z2) + 2] (2:3) It can be checked that the known Mersenne prime indices greater than 31 cannot be p1 p2 expressed as p1 + p2 ¡ 1 for two primes p1; p2 such that both 2 ¡ 1 and 2 ¡ 1 are prime. p1 p2 If z1 6= 0, and 2 ¡ 1 is prime, while z2 is set equal to zero, so that 2 ¡ 1 is allowed to be composite, p1+p2¡1 2 ¡[6((1¡γ1)z1+(1¡γ2)h1+(1¡γ3)h2)+1] = 6(3(h1+z1)h2+γ1h1+γ2z1+γ3h2)+1 (2:4) 0 0 0 0 0 0 for some fractions γ1; γ2; γ3 with 3(h1 +z1)h2 +γ1h1 +γ2z1 +γ3h2 = 6x y §(x +y ); x ; y 2 Z. The Mersenne number 2p1+p2¡1 ¡ 1 is prime if there is no solution to equation (2.4) with γ1 = γ2 = γ3 = 1. If p1 is a given Mersenne prime index, it can be conjectured that p1 ¡ 1 can expressed as the di®erence between two primes p and p2, since an even integer 2N equals p ¡ p2 if 2(N + p2) is given by the sum of the two primes p; p2. The estimated number of prime pairs (p; p + 2N) with p · x is conjectured to be x Y p ¡ 1 ¼ (x) » 2C (2:5) 2N 2 (log x)2 p ¡ 2 p>2 pjN ³ ´ Q 1 where C2 is the twin-prime constant p>2 1 ¡ (p¡1)2 [16][23], and · µ ¶¸ x log log x ¼ (x) · 6:836 C 1 + O (2:6) 2 2 (log x)2 log x Theorem. Every ¯nite even positive integer 2N can be expressed as the di®erence be- tween two primes if the Goldbach conjecture is valid. 2 Proof. For any even integer 2N, N > 2, there exists two primes q1; q2 such that 2N = q1 + q2 if the Goldbach conjecture [7][8] is correct. This equality implies a rela- tion of the form 2N¹ =q ¹1 ¡ q¹2,q ¹1; q¹2 prime, for some N¹ < N, since 2(N ¡ q2) = q1 ¡ q2. It can be assumed that this property holds for all integers 1 · N~ < N¹ and that the lesser prime in each of the di®erences is bounded above by 2N~ ¡ 2. The existence of a prime pair with a di®erence to an arbitrary even integer shall be demonstrated by induction on N¹. 2N¹ + 2 =q ¹1 ¡ q¹2 + 2 = 2N + 2 ¡ 2¹q2 (2:7) = q3 + q4 ¡ 2¹q2 = (q3 ¡ q¹2 + k) ¡ (¹q2 ¡ q4 + k) Since q3 + q4 =q ¹1 +q ¹2 + 2, the indices can be chosen such that q3 > q¹1; q4 < q¹2 + 2 or q¹1 > q3 > q¹2; q4 > q¹2 + 2 unless q3 =q ¹1, q4 =q ¹2 + 2. If the equalities are valid, then 2N¹ ¡2 = q3 ¡q4. Continuing this process with 2N +2 replaced by 2N +2{, { ¸ 2, it follows that the equalities would imply that all even integers less than 2N¹ can be expressed as di®erences between pairs of primes even when arbitrarily large values of N¹ are chosen by 0 subtracting 2q6 from an arbitrarily large integer N = q5 +q6. Consequently, the equalities would imply that every even integer is equal to the di®erence between two primes. Suppose then that the ¯rst set of inequalities holds. Then q3¡q¹2¡1 > 0 andq ¹2¡q4+1 > 0. Since 0 0 2N¹ + 2 = (q3 ¡ q¹2 ¡ k ¡ k ) ¡ (¹q2 ¡ q4 ¡ k ¡ k ) (2:8) 0 0 0 andq ¹2 ¡ q4 ¡ k ¡ k =q ¹2 ¡ k1 ¡ k ¡ (q4 + k2), k1 + k2 = k, k1; k2; k can be chosen such 0 0 that q4 + k2 =q ¹2 ¡ k1 ¡ k ¡ p¹2 withq ¹2 ¡ k1 ¡ k andp ¹2 prime, since q4 + k2 < 2N¹ when 0 q¹2 ¡ q4 ¡ k ¡ k > 0 andq ¹2 < 2N¹. Suppose that the equality between q4 + k2 and the 0 0 0 di®erence between two primes ¯xes k1 + k . Then, k can be adjusted to ¯x q3 ¡ q¹2 ¡ k ¡ k 0 to be prime. It follows that 2N¹ + 2 = q3 ¡ q¹2 ¡ k ¡ k ¡ p¹2 is a di®erence between two primes, withp ¹2 < 2N¹. If q3 < q¹1; q4 > q¹2 + 2, consider the equality 0 0 2N¹ + 2 =q ¹1 ¡ q¹2 + 2 = (¹q1 + q4 ¡ k ¡ k + 2) ¡ (q4 ¡ q¹2 ¡ k ¡ k ) (2:9) 0 0 As q4 ¡ q¹2 ¡ k ¡ k = (q4 ¡ k1 ¡ k ) ¡ (¹q2 ¡ k2), andq ¹2 ¡ k2 is an even integer less than 0 0 2N¹,q ¹2 ¡ k2 = q4 ¡ k1 ¡ k ¡ p¹3 wherep ¹3 is prime and k1 + k is adjusted to render 0 0 0 q4 ¡ k1 ¡ k to be prime. Then k also can be chosen such thatq ¹1 + q4 ¡ k ¡ k + 2 is 0 prime. The lesser prime in the di®erence satis¯es the inequality q4 ¡ q¹2 ¡ k ¡ k < 2N¹ since q4 ¡ q¹2 < q3 ¡ q¹2 < q¹1 ¡ q¹2 = 2N¹.
Recommended publications
  • 1 Mersenne Primes and Perfect Numbers

    1 Mersenne Primes and Perfect Numbers

    1 Mersenne Primes and Perfect Numbers Basic idea: try to construct primes of the form an − 1; a, n ≥ 1. e.g., 21 − 1 = 3 but 24 − 1=3· 5 23 − 1=7 25 − 1=31 26 − 1=63=32 · 7 27 − 1 = 127 211 − 1 = 2047 = (23)(89) 213 − 1 = 8191 Lemma: xn − 1=(x − 1)(xn−1 + xn−2 + ···+ x +1) Corollary:(x − 1)|(xn − 1) So for an − 1tobeprime,weneeda =2. Moreover, if n = md, we can apply the lemma with x = ad.Then (ad − 1)|(an − 1) So we get the following Lemma If an − 1 is a prime, then a =2andn is prime. Definition:AMersenne prime is a prime of the form q =2p − 1,pprime. Question: are they infinitely many Mersenne primes? Best known: The 37th Mersenne prime q is associated to p = 3021377, and this was done in 1998. One expects that p = 6972593 will give the next Mersenne prime; this is close to being proved, but not all the details have been checked. Definition: A positive integer n is perfect iff it equals the sum of all its (positive) divisors <n. Definition: σ(n)= d|n d (divisor function) So u is perfect if n = σ(u) − n, i.e. if σ(u)=2n. Well known example: n =6=1+2+3 Properties of σ: 1. σ(1) = 1 1 2. n is a prime iff σ(n)=n +1 p σ pj p ··· pj pj+1−1 3. If is a prime, ( )=1+ + + = p−1 4. (Exercise) If (n1,n2)=1thenσ(n1)σ(n2)=σ(n1n2) “multiplicativity”.
  • Mersenne and Fermat Numbers 2, 3, 5, 7, 13, 17, 19, 31

    Mersenne and Fermat Numbers 2, 3, 5, 7, 13, 17, 19, 31

    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 ra: 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 ra 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 [l]; see also Kraitchik [4]. 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 [7] and discussed by Uhler [13]. It is clear that 2" —1 can be factored algebraically if ra is composite; hence 2n —1 cannot be prime unless w is prime. Fermat's theorem yields a factor of 2n —1 only when ra + 1 is prime, and hence does not determine any additional cases in which 2"-1 is known to be com- posite. On the other hand, it follows from Euler's criterion that if ra = 0, 3 (mod 4) and 2ra + l is prime, then 2ra + l is a factor of 2n— 1.
  • A Member of the Order of Minims, Was a Philosopher, Mathematician, and Scientist Who Wrote on Music, Mechanics, and Optics

    A Member of the Order of Minims, Was a Philosopher, Mathematician, and Scientist Who Wrote on Music, Mechanics, and Optics

    ON THE FORM OF MERSENNE NUMBERS PREDRAG TERZICH Abstract. We present a theorem concerning the form of Mersenne p numbers , Mp = 2 − 1 , where p > 3 . We also discuss a closed form expression which links prime numbers and natural logarithms . 1. Introduction. The friar Marin Mersenne (1588 − 1648), a member of the order of Minims, was a philosopher, mathematician, and scientist who wrote on music, mechanics, and optics . A Mersenne number is a number of p the form Mp = 2 − 1 where p is an integer , however many authors prefer to define a Mersenne number as a number of the above form but with p restricted to prime values . In this paper we shall use the second definition . Because the tests for Mersenne primes are relatively simple, the record largest known prime number has almost always been a Mersenne prime . 2. A theorem and closed form expression Theorem 2.1. Every Mersenne number Mp , (p > 3) is a number of the form 6 · k · p + 1 for some odd possitive integer k . Proof. In order to prove this we shall use following theorems : Theorem 2.2. The simultaneous congruences n ≡ n1 (mod m1) , n ≡ n2 (mod m2), .... ,n ≡ nk (mod mk) are only solvable when ni = nj (mod gcd(mi; mj)) , for all i and j . i 6= j . The solution is unique modulo lcm(m1; m2; ··· ; mk) . Proof. For proof see [1] Theorem 2.3. If p is a prime number, then for any integer a : ap ≡ a (mod p) . Date: January 4 , 2012. 1 2 PREDRAG TERZICH Proof.
  • The Pseudoprimes to 25 • 109

    The Pseudoprimes to 25 • 109

    MATHEMATICS OF COMPUTATION, VOLUME 35, NUMBER 151 JULY 1980, PAGES 1003-1026 The Pseudoprimes to 25 • 109 By Carl Pomerance, J. L. Selfridge and Samuel S. Wagstaff, Jr. Abstract. The odd composite n < 25 • 10 such that 2n_1 = 1 (mod n) have been determined and their distribution tabulated. We investigate the properties of three special types of pseudoprimes: Euler pseudoprimes, strong pseudoprimes, and Car- michael numbers. The theoretical upper bound and the heuristic lower bound due to Erdös for the counting function of the Carmichael numbers are both sharpened. Several new quick tests for primality are proposed, including some which combine pseudoprimes with Lucas sequences. 1. Introduction. According to Fermat's "Little Theorem", if p is prime and (a, p) = 1, then ap~1 = 1 (mod p). This theorem provides a "test" for primality which is very often correct: Given a large odd integer p, choose some a satisfying 1 <a <p - 1 and compute ap~1 (mod p). If ap~1 pi (mod p), then p is certainly composite. If ap~l = 1 (mod p), then p is probably prime. Odd composite numbers n for which (1) a"_1 = l (mod«) are called pseudoprimes to base a (psp(a)). (For simplicity, a can be any positive in- teger in this definition. We could let a be negative with little additional work. In the last 15 years, some authors have used pseudoprime (base a) to mean any number n > 1 satisfying (1), whether composite or prime.) It is well known that for each base a, there are infinitely many pseudoprimes to base a.
  • Primes and Primality Testing

    Primes and Primality Testing

    Primes and Primality Testing A Technological/Historical Perspective Jennifer Ellis Department of Mathematics and Computer Science What is a prime number? A number p greater than one is prime if and only if the only divisors of p are 1 and p. Examples: 2, 3, 5, and 7 A few larger examples: 71887 524287 65537 2127 1 Primality Testing: Origins Eratosthenes: Developed “sieve” method 276-194 B.C. Nicknamed Beta – “second place” in many different academic disciplines Also made contributions to www-history.mcs.st- geometry, approximation of andrews.ac.uk/PictDisplay/Eratosthenes.html the Earth’s circumference Sieve of Eratosthenes 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 Sieve of Eratosthenes We only need to “sieve” the multiples of numbers less than 10. Why? (10)(10)=100 (p)(q)<=100 Consider pq where p>10. Then for pq <=100, q must be less than 10. By sieving all the multiples of numbers less than 10 (here, multiples of q), we have removed all composite numbers less than 100.
  • Arxiv:1412.5226V1 [Math.NT] 16 Dec 2014 Hoe 11

    Arxiv:1412.5226V1 [Math.NT] 16 Dec 2014 Hoe 11

    q-PSEUDOPRIMALITY: A NATURAL GENERALIZATION OF STRONG PSEUDOPRIMALITY JOHN H. CASTILLO, GILBERTO GARC´IA-PULGAR´IN, AND JUAN MIGUEL VELASQUEZ-SOTO´ Abstract. In this work we present a natural generalization of strong pseudoprime to base b, which we have called q-pseudoprime to base b. It allows us to present another way to define a Midy’s number to base b (overpseudoprime to base b). Besides, we count the bases b such that N is a q-probable prime base b and those ones such that N is a Midy’s number to base b. Furthemore, we prove that there is not a concept analogous to Carmichael numbers to q-probable prime to base b as with the concept of strong pseudoprimes to base b. 1. Introduction Recently, Grau et al. [7] gave a generalization of Pocklignton’s Theorem (also known as Proth’s Theorem) and Miller-Rabin primality test, it takes as reference some works of Berrizbeitia, [1, 2], where it is presented an extension to the concept of strong pseudoprime, called ω-primes. As Grau et al. said it is right, but its application is not too good because it is needed m-th primitive roots of unity, see [7, 12]. In [7], it is defined when an integer N is a p-strong probable prime base a, for p a prime divisor of N −1 and gcd(a, N) = 1. In a reading of that paper, we discovered that if a number N is a p-strong probable prime to base 2 for each p prime divisor of N − 1, it is actually a Midy’s number or a overpseu- doprime number to base 2.
  • Appendix a Tables of Fermat Numbers and Their Prime Factors

    Appendix a Tables of Fermat Numbers and Their Prime Factors

    Appendix A Tables of Fermat Numbers and Their Prime Factors The problem of distinguishing prime numbers from composite numbers and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic. Carl Friedrich Gauss Disquisitiones arithmeticae, Sec. 329 Fermat Numbers Fo =3, FI =5, F2 =17, F3 =257, F4 =65537, F5 =4294967297, F6 =18446744073709551617, F7 =340282366920938463463374607431768211457, Fs =115792089237316195423570985008687907853 269984665640564039457584007913129639937, Fg =134078079299425970995740249982058461274 793658205923933777235614437217640300735 469768018742981669034276900318581864860 50853753882811946569946433649006084097, FlO =179769313486231590772930519078902473361 797697894230657273430081157732675805500 963132708477322407536021120113879871393 357658789768814416622492847430639474124 377767893424865485276302219601246094119 453082952085005768838150682342462881473 913110540827237163350510684586298239947 245938479716304835356329624224137217. The only known Fermat primes are Fo, ... , F4 • 208 17 lectures on Fermat numbers Completely Factored Composite Fermat Numbers m prime factor year discoverer 5 641 1732 Euler 5 6700417 1732 Euler 6 274177 1855 Clausen 6 67280421310721* 1855 Clausen 7 59649589127497217 1970 Morrison, Brillhart 7 5704689200685129054721 1970 Morrison, Brillhart 8 1238926361552897 1980 Brent, Pollard 8 p**62 1980 Brent, Pollard 9 2424833 1903 Western 9 P49 1990 Lenstra, Lenstra, Jr., Manasse, Pollard 9 p***99 1990 Lenstra, Lenstra, Jr., Manasse, Pollard
  • The Distribution of Pseudoprimes

    The Distribution of Pseudoprimes

    The Distribution of Pseudoprimes Matt Green September, 2002 The RSA crypto-system relies on the availability of very large prime num- bers. Tests for finding these large primes can be categorized as deterministic and probabilistic. Deterministic tests, such as the Sieve of Erastothenes, of- fer a 100 percent assurance that the number tested and passed as prime is actually prime. Despite their refusal to err, deterministic tests are generally not used to find large primes because they are very slow (with the exception of a recently discovered polynomial time algorithm). Probabilistic tests offer a much quicker way to find large primes with only a negligibal amount of error. I have studied a simple, and comparatively to other probabilistic tests, error prone test based on Fermat’s little theorem. Fermat’s little theorem states that if b is a natural number and n a prime then bn−1 ≡ 1 (mod n). To test a number n for primality we pick any natural number less than n and greater than 1 and first see if it is relatively prime to n. If it is, then we use this number as a base b and see if Fermat’s little theorem holds for n and b. If the theorem doesn’t hold, then we know that n is not prime. If the theorem does hold then we can accept n as prime right now, leaving a large chance that we have made an error, or we can test again with a different base. Chances improve that n is prime with every base that passes but for really big numbers it is cumbersome to test all possible bases.
  • Cullen Numbers with the Lehmer Property

    Cullen Numbers with the Lehmer Property

    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000–000 S 0002-9939(XX)0000-0 CULLEN NUMBERS WITH THE LEHMER PROPERTY JOSE´ MAR´IA GRAU RIBAS AND FLORIAN LUCA Abstract. Here, we show that there is no positive integer n such that n the nth Cullen number Cn = n2 + 1 has the property that it is com- posite but φ(Cn) | Cn − 1. 1. Introduction n A Cullen number is a number of the form Cn = n2 + 1 for some n ≥ 1. They attracted attention of researchers since it seems that it is hard to find primes of this form. Indeed, Hooley [8] showed that for most n the number Cn is composite. For more about testing Cn for primality, see [3] and [6]. For an integer a > 1, a pseudoprime to base a is a compositive positive integer m such that am ≡ a (mod m). Pseudoprime Cullen numbers have also been studied. For example, in [12] it is shown that for most n, Cn is not a base a-pseudoprime. Some computer searchers up to several millions did not turn up any pseudo-prime Cn to any base. Thus, it would seem that Cullen numbers which are pseudoprimes are very scarce. A Carmichael number is a positive integer m which is a base a pseudoprime for any a. A composite integer m is called a Lehmer number if φ(m) | m − 1, where φ(m) is the Euler function of m. Lehmer numbers are Carmichael numbers; hence, pseudoprimes in every base. No Lehmer number is known, although it is known that there are no Lehmer numbers in certain sequences, such as the Fibonacci sequence (see [9]), or the sequence of repunits in base g for any g ∈ [2, 1000] (see [4]).
  • Some New Results on Odd Perfect Numbers

    Some New Results on Odd Perfect Numbers

    Pacific Journal of Mathematics SOME NEW RESULTS ON ODD PERFECT NUMBERS G. G. DANDAPAT,JOHN L. HUNSUCKER AND CARL POMERANCE Vol. 57, No. 2 February 1975 PACIFIC JOURNAL OF MATHEMATICS Vol. 57, No. 2, 1975 SOME NEW RESULTS ON ODD PERFECT NUMBERS G. G. DANDAPAT, J. L. HUNSUCKER AND CARL POMERANCE If ra is a multiply perfect number (σ(m) = tm for some integer ί), we ask if there is a prime p with m = pan, (pa, n) = 1, σ(n) = pα, and σ(pa) = tn. We prove that the only multiply perfect numbers with this property are the even perfect numbers and 672. Hence we settle a problem raised by Suryanarayana who asked if odd perfect numbers neces- sarily had such a prime factor. The methods of the proof allow us also to say something about odd solutions to the equation σ(σ(n)) ~ 2n. 1* Introduction* In this paper we answer a question on odd perfect numbers posed by Suryanarayana [17]. It is known that if m is an odd perfect number, then m = pak2 where p is a prime, p Jf k, and p = a z= 1 (mod 4). Suryanarayana asked if it necessarily followed that (1) σ(k2) = pa , σ(pa) = 2k2 . Here, σ is the sum of the divisors function. We answer this question in the negative by showing that no odd perfect number satisfies (1). We actually consider a more general question. If m is multiply perfect (σ(m) = tm for some integer t), we say m has property S if there is a prime p with m = pan, (pa, n) = 1, and the equations (2) σ(n) = pa , σ(pa) = tn hold.
  • Generation of Pseudoprimes Section 1:Introduction

    Generation of Pseudoprimes Section 1:Introduction

    Generation of Pseudoprimes Danielle Stewart Swenson College of Science and Engineering University of Minnesota [email protected] Section 1:Introduction: Number theory is a branch of mathematics that looks at the many properties of integers. The properties that are looked at in this paper are specifically related to pseudoprime numbers. Positive integers can be partitioned into three distinct sets. The unity, composites, and primes. It is much easier to prove that an integer is composite compared to proving primality. Fermat’s Little Theorem p If p is prime and a is any integer, then a − a is divisible by p. This theorem is commonly used to determine if an integer is composite. If a number does not pass this test, it is shown that the number must be composite. On the other hand, if a number passes this test, it does not prove this integer is prime (Anderson & Bell, 1997) . An example 5 would be to let p = 5 and let a = 2. Then 2 − 2 = 32 − 2 = 30 which is divisible by 5. Since p = 5 5 is prime, we can choose any a as a positive integer and a − a is divisible by 5. Now let p = 4 and 4 a = 2 . Then 2 − 2 = 14 which is not divisible by 4. Using this theorem, we can quickly see if a number fails, then it must be composite, but if it does not fail the test we cannot say that it is prime. This is where pseudoprimes come into play. If we know that a number n is composite but n n divides a − a for some positive integer a, we call n a pseudoprime.
  • On Types of Elliptic Pseudoprimes

    On Types of Elliptic Pseudoprimes

    journal of Groups, Complexity, Cryptology Volume 13, Issue 1, 2021, pp. 1:1–1:33 Submitted Jan. 07, 2019 https://gcc.episciences.org/ Published Feb. 09, 2021 ON TYPES OF ELLIPTIC PSEUDOPRIMES LILJANA BABINKOSTOVA, A. HERNANDEZ-ESPIET,´ AND H. Y. KIM Boise State University e-mail address: [email protected] Rutgers University e-mail address: [email protected] University of Wisconsin-Madison e-mail address: [email protected] Abstract. We generalize Silverman's [31] notions of elliptic pseudoprimes and elliptic Carmichael numbers to analogues of Euler-Jacobi and strong pseudoprimes. We inspect the relationships among Euler elliptic Carmichael numbers, strong elliptic Carmichael numbers, products of anomalous primes and elliptic Korselt numbers of Type I, the former two of which we introduce and the latter two of which were introduced by Mazur [21] and Silverman [31] respectively. In particular, we expand upon the work of Babinkostova et al. [3] on the density of certain elliptic Korselt numbers of Type I which are products of anomalous primes, proving a conjecture stated in [3]. 1. Introduction The problem of efficiently distinguishing the prime numbers from the composite numbers has been a fundamental problem for a long time. One of the first primality tests in modern number theory came from Fermat Little Theorem: if p is a prime number and a is an integer not divisible by p, then ap−1 ≡ 1 (mod p). The original notion of a pseudoprime (sometimes called a Fermat pseudoprime) involves counterexamples to the converse of this theorem. A pseudoprime to the base a is a composite number N such aN−1 ≡ 1 mod N.