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Characterizations of Prime K−Tuples Using Binomial Expressions International Mathematical Forum, Vol. 6, 2011, no. 44, 2165 - 2168 Characterizations of Prime k−Tuples using Binomial Expressions Zvi Retchkiman K¨onigsberg Instituto Polit´ecnico Nacional CIC Mineria 17-2, Col. Escandon Mexico D.F 11800, Mexico [email protected] Abstract In number theory, a prime k-tuple is an ordered set of values (i.e. a vector) representing a repeatable pattern of prime numbers. Some of the most common k-tuples are: twin primes, cousin primes, sexy primes, prime triplets and prime quadruplets. This paper gives charac- terizations of k-tuple primes in terms of binomial coefficients. Departing from a known characterization of prime numbers in terms of binomial coefficients, necessary and sufficient conditions for higher order tuples in terms of binomial coefficients are constructed. Mathematics Subject Classification: 11A41, 11B65 Keywords: Prime k-tuple, Binomial coefficients 1 Introduction In number theory, a prime k-tuple is an ordered set of values (i.e. a vector) representing a repeatable pattern of prime numbers, being some of the most common: • Twin primes, a set of two prime numbers that differ by two except for the pair (2, 3). • Cousin primes, a set of two prime numbers that differ by four. • Sexy primes, a set of pairs of prime numbers that differ by six • Prime triplets, a set of three prime numbers of the form (p, p + 2, p + 6) or (p, p + 4, p + 6) with the exceptions of (2, 3, 5) and (3, 5, 7). 2166 Z. Retchkiman K¨onigsberg • Prime quadruplets, a set of four primes of the form (p, p+2, p+6, p+8). Notice that a prime triplet contains a pair of twin primes (p and p + 2, or p + 4 and p + 6), a pair of cousin primes (p and p + 4, or p + 2 and p + 6), and a pair of sexy primes (p and p + 6) and also that a prime quadruplet contains two pairs of twin primes and two overlapping prime triplets. In this paper characterizations of k-tuple primes in terms of binomial coeffi- cients are presented. Departing from a known characterization of prime num- bers in terms of binomial coefficients [1], necessary and sufficient conditions for higher order tuples in terms of binomial coefficients are constructed. It is worth mentioning that in [2] the twin primes characterization in terms of binomial coefficients is given, however its proof is based on generating func- tions which is distinct to the argument provided here to prove it. The paper is organized as follows. Section 1, presents the main results of this work ending with section 2, where some concluding remarks are given. 2 Characterizations of k-tuple primes In this section, the problem of giving necessary and sufficient conditions for k-tuple primes in terms of binomial coefficients is addressed. Theorem 1 [1] Let 2n +1 be any number with n ≥ 2 and consider the set ΠT = {p : p is a prime such that 3 ≤ p<2n +1}. Then the number 2n +1 is n p−1 + 2 a prime number if and only if ∀pεΠT p | p − 1 Theorem 2 Let 2n − 1 and 2n +1 be any two numbers with n ≥ 3 and consider the set ΠT = {p : p is a prime such that 3 ≤ p<2n −1}. Then the n p−3 + 2 pair (2n−1, 2n+1) is a twin prime pair if and only if ∀pεΠT p | . p − 2 Proof. First, assume that the pair (2n − 1, 2n + 1) is a twin prime pair we n p−3 ∀pε ,p | + 2 must show that Π p − . Since each one of them is a prime 2 n + p−3 n + p−1 number, from theorem 1: p | 2 and p |− 2 but p − 1 p − 1 n + p−3 n + p−1 n + p−3 2 = 2 − 2 > 0 (1) p − 2 p − 1 p − 1 n + p−3 Therefore p | 2 as desired. p − 2 Characterizations of prime k−tuples 2167 Now let us prove the converse. Assume that the pair (2n − 1, 2n + 1) is not a twin prime. If p | (2n − 1) ⇒ 2n − 1=p(2m + 1) with m ≥ 1. Then n p−3 p − mp + 2 = 1+ and the binomial coefficient becomes (p − 1+mp)(p − 2+mp) ···(2 + mp) (p − 2)! In case p | (2n + 1) we get that (p − 2+mp)(p − 1+mp) ···(1 + mp) (p − 2)! n + p−3 In either case p does not divide 2 . p − 2 Remark 3 The characterization given in theorem 2 for twin primes is ex- pressed in terms of one binomial coefficient thanks to the recursive formula for binomial coefficients (1) applied to the binomial coefficient conditions for each one of the primes 2n − 1 and 2n +1 (this can be interpreted as a linear combination with coefficients equal to one). Unfortunately, it is not possible to obtain a similar recursive formula when the difference between the pair of primes is higher than two (as can be proved by direct computation). This will be seen in the next results whose proofs are obtained straightforwardly by applying theorems 1 and 2. Theorem 4 Let 2n+1 and 2n+5 be any two numbers with n ≥ 2 and consider {p p ≤ p< n } {p p the sets ΠT1 = : is a prime such that 3 2 +1 and ΠT2 = : is a prime such that 3 ≤ p<2n+5}. Then the pair (2n+1, 2n+5)is a cousin prime n p−1 n p+3 + 2 + 2 pair if and only if ∀pεΠT p | and ∀pεΠT p | 1 p − 1 2 p − 1 Theorem 5 Let 2n+1 and 2n+7 be any two numbers with n ≥ 2 and consider {p p ≤ p< n } {p p the sets ΠT1 = : is a prime such that 3 2 +1 and ΠT2 = : is a prime such that 3 ≤ p<2n+7}. Then the pair (2n+1, 2n+7) is a sexy prime n p−1 n p+5 + 2 + 2 pair if and only if ∀pεΠT p | and ∀pεΠT p | 1 p − 1 2 p − 1 Theorem 6 Let 2n − 1, 2n +1, 2n +3 and 2n +5 be any four numbers n ≥ {p p ≤ with 3 and consider the sets ΠT1 = : is a prime such that 3 p< n − } {p p ≤ p< n } 2 1 , ΠT2 = : is a prime such that 3 2 +3 . Then (2n − 1, 2n +1, 2n +5) or (2n −1, 2n +3, 2n +5) are a prime triplet if and n p−3 n p+3 ∀pε p | + 2 ∀pε p | + 2 ∀pε only if ΠT1 p − and ΠT2 p − or ΠT1 2 1 n p−3 n p+1 + 2 + 2 p | and ∀pεΠT p | . p − 1 2 p − 2 2168 Z. Retchkiman K¨onigsberg Theorem 7 Let 2n − 1, 2n +1, 2n +5 and 2n +7 be any four numbers with n ≥ {p p ≤ p< 3 and consider the sets ΠT1 = : is a prime such that 3 n − } {p p ≤ p< n } 2 1 and ΠT2 = : is a prime such that 3 2 +5 . Then n − , n , n , n ∀pε (2 1 2 +12 +5 2 +7) is a prime quadruplet if and only if ΠT1 n p−3 n p+3 + 2 + 2 p | and ∀pεΠT p | p − 2 2 p − 2 Remark 8 Notice that the conditions given in theorem 7 are not unique since a prime quadruplet can be considered as two pairs of twin primes (as was done here), or as two overlapping prime triplets. This means, that theorem 7 could have been stated in terms of prime triplets. 3 Conclusions In this paper characterizations of k-tuple primes in terms of binomial coeffi- cients were presented. These were implemented in a PC software where they were tested in numerical cases showing to work properly. References [1] J. D’Angelo, Number theoretic properties of certain CR mappings, J. Geom. Anal. 14 (2004) 215-229. [2] K. Dilcher and K. Stolarsky, A Pascal-Type Triangle Charactrizing Twin Primes, (T he Mathematical Association of America Monthly), 673:681, 2005. Received: March, 2011.
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