On a sequence of prime numbers
... primes, or it excludes all non-occurring primes. If the latter is true there is a finite decision procedure for occurrence or non-occurrence of a given prime q, and the set of primes generated is recursive. It seems likely that a) an infinite set of primes do not occur in (1) and b) all absent prime ...
... primes, or it excludes all non-occurring primes. If the latter is true there is a finite decision procedure for occurrence or non-occurrence of a given prime q, and the set of primes generated is recursive. It seems likely that a) an infinite set of primes do not occur in (1) and b) all absent prime ...
Primes and Modular Arithmetic
... “fingerprint” mod each of the prime values. In other words, any number k (up to the product of all the primes) can be recognized by its “set of remainders”. ...
... “fingerprint” mod each of the prime values. In other words, any number k (up to the product of all the primes) can be recognized by its “set of remainders”. ...
DECIMAL EXPANSION OF 1/P AND SUBGROUP SUMS
... The computation of s(p, l) for any prime p and any divisor l of p − 1 is equivalent to the computation of the sum U1 + · · · + Ul where 1/p is expressed in base b for a primitive root b (mod p). In particular, the question arises as to whether s(p, l) equals p for any l > 3 at all? We shall now show ...
... The computation of s(p, l) for any prime p and any divisor l of p − 1 is equivalent to the computation of the sum U1 + · · · + Ul where 1/p is expressed in base b for a primitive root b (mod p). In particular, the question arises as to whether s(p, l) equals p for any l > 3 at all? We shall now show ...
A New Fibonacci-like Sequence of Composite Numbers
... The aim of this note is to provide a new Fibonacci-like sequence with even smaller initial values. We prove Theorem 1. Let {An }, n ≥ 0, be defined by (1)–(2) with the following relatively prime initial values: a = 106276436867, b = 35256392432. Then {An } contains no prime number. Proof. The idea o ...
... The aim of this note is to provide a new Fibonacci-like sequence with even smaller initial values. We prove Theorem 1. Let {An }, n ≥ 0, be defined by (1)–(2) with the following relatively prime initial values: a = 106276436867, b = 35256392432. Then {An } contains no prime number. Proof. The idea o ...
77 Seventy-Seven LXXVII
... The number 77, and its sibling 49, are the only two-digit numbers whose home prime is not known. The home prime is obtained by repeatedly taking a number and concatenating its prime factors until you reach a prime. The first six terms of the sequence starting at 49 are 49, 77, 711, 3379, 31109, 1323 ...
... The number 77, and its sibling 49, are the only two-digit numbers whose home prime is not known. The home prime is obtained by repeatedly taking a number and concatenating its prime factors until you reach a prime. The first six terms of the sequence starting at 49 are 49, 77, 711, 3379, 31109, 1323 ...
21 Twenty-One XXI
... The number 21 has four divisors: 1, 3, 7, 21. The number 21 is the seventeenth deficient number: s(21) = 1 + 3 + 7 = 11 < 21. As the sum of four or fewer squares: 21 = 12 + 22 + 42 = 22 + 22 + 22 + 32 . As the sum of nine or fewer cubes: 21 = 5 13 + 2 23 . As a difference of two squares: 21 = 52 ...
... The number 21 has four divisors: 1, 3, 7, 21. The number 21 is the seventeenth deficient number: s(21) = 1 + 3 + 7 = 11 < 21. As the sum of four or fewer squares: 21 = 12 + 22 + 42 = 22 + 22 + 22 + 32 . As the sum of nine or fewer cubes: 21 = 5 13 + 2 23 . As a difference of two squares: 21 = 52 ...
Chapter 3: Primes and their Distribution
... How does this simplify the test for composite numbers? Well for testing whether 983 is prime or composite we only need to see if the primes less than or equal to 983 31 go into 983. The primes 31 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 and 31 There are 11 primes 31 and we only need to see ...
... How does this simplify the test for composite numbers? Well for testing whether 983 is prime or composite we only need to see if the primes less than or equal to 983 31 go into 983. The primes 31 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 and 31 There are 11 primes 31 and we only need to see ...