Results 1 
7 of
7
DIVISORS OF SHIFTED PRIMES
"... Abstract. We bound from below the number of shifted primes p+s ≤ x that have a divisor in a given interval (y, z]. Kevin Ford has obtained upper bounds of the expected order of magnitude on this quantity as well as lower bounds in a special case of the parameters y and z. We supply here the correspo ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
Abstract. We bound from below the number of shifted primes p+s ≤ x that have a divisor in a given interval (y, z]. Kevin Ford has obtained upper bounds of the expected order of magnitude on this quantity as well as lower bounds in a special case of the parameters y and z. We supply here the corresponding lower bounds in a broad range of the parameters y and z. As expected, these bounds depend heavily on our knowledge about primes in arithmetic progressions. As an application of these bounds, we determine the number of shifted primes that appear in a multiplication table up to multiplicative constants. 1.
Carmichael numbers and pseudoprimes Notes by G.J.O. Jameson
"... Recall that Fermat’s “little theorem ” says that if p is prime and a is not a multiple of p, then ap−1 ≡ 1 mod p. This theorem gives a possible way to detect primes, or more exactly, nonprimes: if for a certain a coprime to n, an−1 is not congruent to 1 mod n, then, by the theorem, n is not ..."
Abstract
 Add to MetaCart
Recall that Fermat’s “little theorem ” says that if p is prime and a is not a multiple of p, then ap−1 ≡ 1 mod p. This theorem gives a possible way to detect primes, or more exactly, nonprimes: if for a certain a coprime to n, an−1 is not congruent to 1 mod n, then, by the theorem, n is not
Notes by G.J.O. Jameson
"... Recall that Fermat’s “little theorem ” says that if p is prime and a is not a multiple of p, then ap−1 ≡ 1 mod p. This theorem gives a possible way to detect primes, or more exactly, nonprimes: if for some positive a ≤ n − 1, an−1 is not congruent to 1 mod n, then, by the theorem, n is ..."
Abstract
 Add to MetaCart
Recall that Fermat’s “little theorem ” says that if p is prime and a is not a multiple of p, then ap−1 ≡ 1 mod p. This theorem gives a possible way to detect primes, or more exactly, nonprimes: if for some positive a ≤ n − 1, an−1 is not congruent to 1 mod n, then, by the theorem, n is
GEOMETRIC PROPERTIES OF POINTS ON MODULAR HYPERBOLAS
"... ABSTRACT. Given an integer n � 2, let Hn be the set Hn = {(a, b) : ab ≡ 1 (mod n), 1 � a, b � n − 1} and let M(n) be the maximal difference of b − a for (a, b) ∈ Hn. We prove that for almost all n, n − M(n) = O ( n 1/2+o(1)). We also improve some previously known upper and lower bounds on the numb ..."
Abstract
 Add to MetaCart
ABSTRACT. Given an integer n � 2, let Hn be the set Hn = {(a, b) : ab ≡ 1 (mod n), 1 � a, b � n − 1} and let M(n) be the maximal difference of b − a for (a, b) ∈ Hn. We prove that for almost all n, n − M(n) = O ( n 1/2+o(1)). We also improve some previously known upper and lower bounds on the number of vertices of the convex closure of Hn. 1.
J. Aust. Math. Soc. 94 (2013), 268–275 doi:10.1017/S1446788712000547 CARMICHAEL NUMBERS IN ARITHMETIC PROGRESSIONS
, 2013
"... We prove that when (a, m) = 1 and a is a quadratic residue mod m, there are infinitely many Carmichael numbers in the arithmetic progression a mod m. Indeed the number of them up to x is at least x 1/5 when x is large enough (depending on m). ..."
Abstract
 Add to MetaCart
We prove that when (a, m) = 1 and a is a quadratic residue mod m, there are infinitely many Carmichael numbers in the arithmetic progression a mod m. Indeed the number of them up to x is at least x 1/5 when x is large enough (depending on m).