Combinatorial Primality Test

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1 Cobinatorial Priality Test Maheswara Rao Valluri School of Matheatical and Coputing Sciences Fiji National University, Derrick Capus, Suva, Fiji E-ail: Abstract This paper provides proofs of ( the results of Laisant - Beaujeux: (1 If an integer of the for n = 4k + 1, k > 0 is prie, then 1(od n, =, and ( If an integer of the ( for n = 4k +3, k 0 is prie, then 1(od n, =. In addition, the author proposes iportant conjectures based on the converse of the above theores which ai to establish priality of n. These conjectures are scrutinized by the given cobinatorial priality test algorith which can also distinguish patterns of prie n whether it is of the for 4k + 1 or 4k Introduction A positive integer n is to be called a prie if it has only divisor of 1 and itself, otherwise n is called a coposite. There are, by Euclid s theore (about 350BC infinitely any pries. Pries are ainly categorized into three patterns of the for 4k + 1, 4k + and 4k + 3. The integer is only the even prie that is of the for 4k +. Pries of the for 4k + 1 are 5, 13, 17, 9, 37, 41,...etc., and of the for 4k + 3 are 3, 7, 11, 19, 3, 31, 43,...etc. There are also any other patterns of pries which are listed on the Online Encyclopedia Integer Sequence (OEIS. Readers are referred to search for any prie pattern on the OEIS [9]. Pries are building blocks for any coposite. A coposite is coposed by pries in soe order. For instance, 143 is coposed by pries, 11 and 13. It is clear that one can easily copose any coposite by ultiplication of pries but decoposition of the coposite is a challenging coputational proble, so called the integer factorization proble. The RSA cryptosyste [11] was constructed based on the integer factorization proble. This is an NP-interediate proble on a classical coputer. However, this proble can be reduced to a polynoial tie proble on a quantu coputer due to Shor s algorith [1]. In 1808, a French atheatician, Christian Krap introduced the notation! for factorials. The factorial of n is the product of all positive integers less than or equal to n. In Krap s notation, n! = n((n !. By convention, one defines 0! = 1! = 1. Suppose that a set S contains n distinct objects. Then, one obtains that there are exactly n! perutations of S. If there are r objects chosen fro the set S that contains n objects, this is equal to Wilson s theore [6, Theore 80] which is based on factorials. ( n r n! =. We now recall the (n r!r! Theore 1. (Wilson s Theore An integer n is prie if and only if (! 1(od n. 1

2 The Wilson s theore is not helpful to eploy in practice, since the Wilson s priality test requires O(n operations to copute it. Later, a variant of the Wilson s theore was discovered in 1961 [8] which is stated as follows: Lea 1. Let n = 4k + 1 be a prie. Then (! ( 1v (od n, where v is the nuber of quadratic non-residue less than 1 n. We recall the Ferat s Little Theore and Ferat-Euler Theore [3] which are used to prove the ain theores in the next section. Theore. (Ferat s Little Theore Let n be a prie and integer a is a priitive such that gcd(a, n = 1. Then, a 1(od n. The Ferat s Little Theore is a probablistic priality test. The origin of Carichael nubers is based on the cases where the Ferat s Little Theore fails []. However, the Ferat s Little Theore becae a basis to establish any priality tests. There is a special case of the Ferat s Little Theore which is given as follows: Theore 3. (Ferat-Euler Theore If gcd(a, n = 1, then a φ(n 1(od n. Reark 1. For n is a prie, we have φ(n =. The purpose of the paper is to provide proofs of the results of ( Laisant-Beaujeux [4, page 77]: (1 If an integer of the for n = 4k + 1, k > 0 is prie, then 1(od n, =. ( ( If an integer of the for n = 4k + 3, k 0 is prie, then 1(od n, =. Furtherore, this paper conjectures stateents based on the converse of the above theores, and also presents an algorith for cobinatorial priality test. This test requires O( ( operations for its priality. Main Results This section presents proofs of the results of Laisant - Beaujeux [4, page 77]. ( Theore 4. If an integer of the for n = 4k + 1, k > 0 is prie, then 1(od n, = Ṗroof. Suppose that n is prie of the for 4k + 1. Then, n has a priitive root. Let a be a priitive root odulo n. Then the integers 1, a, a,..., a n are congruence odulo n, to soe order, 1,, 3,...,. Hence, we have (! 1 a a a n (od n a (n ( (od n. Consequently, we have! a ((n+1 8 (od n. And, ( (! [( a (n ( a ((n 5!] a ((n+1 4 (od n. 4 To analyze further the expression, we eploy n = 4k + 1, where k is a natural nuber. As k < 4k =, then a k 1(od n. However, a 4k a 1(od n, by Ferat s Little Theore. As (a k 4 a 4k 1(od n, then a k ±1(od n, So a k 1(od n. Hence, ( a ((n 5 4 a (4k+1 1(4k (a k 4k 4 ( 1 4(k 1 1(od n.

3 Exaple 1. For prie n = 17, 11 1(od 7. ( (od 17 and for coposite n = 7, Theore 5. If an integer of the for n = 4k + 3, k 0 is prie, then Ṗroof. Extracting fro the above proof, ( a ((n 5 4 (od n ( ( (od n, = also holds for this case. To analyze the expression, we use n = 4k + 3, where k is a natural nuber. As k < (k + 1 =, then a k 1(od n. However, a (k+1 a 1(od n, by Ferat s Little Theore. As a 4k+ a ( 1(od n, then a k+1 1(od n. Hence, ( a (n ( 4 a (4k+3 1(4k a (4k+(4k 4 a (k+1(k 1 ( 1 (k 1 1(od n. Exaple. For n = 31, ( (od 31. Reark. Algorith 1 checks the validity of the converse of the theore 4 and 5. Thus, the author proposes Conjectures 1 and. ( Conjecture 1. If 1(od n, =, then n = 4k + 1 is a prie. ( Conjecture. If 1(od n, =, then n = 4k + 3 is a prie. Reark 3. Through the assistance of Oliver Knill, it was found that the Conjecture fails at n = Note that the coposite nubers that satisfy the Conjecture are cobinatorial pseudopries. These nubers could be rarer than Carichael nubers. Furtherore, the author proposes the following conjecture. Conjecture 3. There are infinitely any cobinatorial pseudopries. 3 Cobinatorial Priality Test An iportant challenge in nuber theory is to efficiently deterine whether an integer n is prie or coposite. Miller-Rabin Priality Test [7,10] and Elliptic Curve Priality Test [5] are well known efficient priality tests. However, these are probabilistic priality tests. In 00, Agrawal et al., discovered an unconditional deterinistic priality test [1] which requires O((log n 6+ɛ steps on a classical coputer. The Wilson s priality test is also a deterinistic test but it is not coputationally efficient. In the algorith 1, we provide a priality test based on cobinatorics which requires O( ( steps in order to test whether n is prie or coposite. 3

4 Algorith 1 Algorith for Cobinatorial Priality Test Input : An integer n Runtie : O( ( operations Procedure :( 1: Copute r(od n, = : if r = 1, then 3: Declare n is prie of the for 4k + 1 4: else if r = 1, then 5: Declare n is prie of the for 4k + 3 6: else 7: Declare n is coposite. 8: end if 9:end if Output : Declare whether the integer n is prie or coposite 4 Conclusion This paper has provided proofs of the results of Laisant - Beaujeux on priality and proposed corresponding conjectures based on their converse stateents. Furtherore, the paper has presented an algorith for cobinatorial priality test which checks the validity of the conjectures. The test requires O( ( operations to test whether n is prie or coposite. It is also noted that this test could also distinguish patterns of prie whether it is of the for 4k + 1 or 4k + 3. Acknowledgeent The author of the paper thanks Oliver Knill for his coents and suggestions on the Conjecture. References [1] Agrawal, M., Kayal, N., and Saxena, N., PRIMES is in P, Ann. of Math., 160(, (004. [] Carichael, R.D. Note on a nuber theory function. Bull. Aer. Math. Soc., 16:3 38, [3] Crandall, R., and Poerance, C., Prie Nubers: A Coputational Perspective, Springer- Verlag, New York (001. [4] Dickson, L.E., History of the theory of nubers. Vol. I: Divisibility and priality, New York: Dover Publications, ISBN , MR , Zbl (1919. [5] Goldwasser, S.; Kilian, J. Priality Testing Using Elliptic Curves, Journal of the ACM 46(4, (1999. MR18117 (00e:1118. [6] Hardy, G.H., and Wright, E.M., An Introduction to the Theory of Nubers, Oxford University Press, 4th Edition, (

5 [7] Miller, G.L., Rieann s hypothesis and tests for priality, Journal of Coputer and Syste Sciences, 13 (3: (1976. [8] Mordell, L. J., The congruence (p 1/! ±1(od p. Aer. Math. Monthly 68, (1961. [9] Online Encyclopedia Integer Sequence, [10] Rabin, M.O., Probabilistic algoriths for testing priality, J. Nuber Theory 1, (1980. [11] Rivest, R., Shair, A, and Adlean, L., A ethod for obtaining digital signature and public-key cryptosyste, Counication of the ACM, 1(, (1978. [1] Shor, P.W., Polynoial-Tie Algoriths for Prie Factorization and Discrete Logariths on a Quantu Coputer, SIAM Journal on Coputing, 6(5, (

Congruences involving Bernoulli and Euler numbers Zhi-Hong Sun

The aer will aear in Journal of Nuber Theory. Congruences involving Bernoulli Euler nubers Zhi-Hong Sun Deartent of Matheatics, Huaiyin Teachers College, Huaian, Jiangsu 300, PR China Received January