A234001 Lowest common modulus to which the set of residue classes (mod 4n) that all the primes represented by a certain quadratic form of discriminant = -4n belong to, can be simplified to, for all quadratic forms of discriminant = -4n.

4, 8, 3, 4, 20, 24, 14, 8, 12, 40, 11, 12, 52, 56, 30, 8, 68, 24, 19, 20, 84, 88, 46, 24, 20, 104, 3, 28, 116, 120, 62, 8, 132, 136, 35, 12, 148, 152, 78, 40, 164, 168, 43, 44, 60, 184, 94, 24, 28, 40, 51, 52, 212, 24, 110, 56, 228, 232, 59, 60, 244, 248, 42, 8, 260, 264, 67, 68, 276, 280

Offset: 1

Author

V. Raman, Dec 18 2013

Keywords

• nonn
• uned

Comments

If n is a convenient number (A000926), the set of residue classes (mod 4n) that a prime p represented by x^2+n*y^2 belong to are those for which p is a quadratic residue (mod 4n) or p-n is a quadratic residue (mod 4n), assuming that p^2 does not divide n. For non-convenient numbers n, some of the primes in these set of residue classes (mod 4n) can be represented by x^2+n*y^2, but not all.

A prime p such that p^2 does not divide n, can be represented by some primitive quadratic form of discriminant = -4n, if and only if -n is a quadratic residue (mod p).

A prime p can be represented by some quadratic form of discriminant = -4n if and only if there is a multiple of p that can be written in the x^2+n*y^2 form, in which prime factor of p appears raised to an odd power or if p = 2 and n == 3 (mod 4).

a(n) is always a divisor of 4n.

If n is squarefree and n == 1 (mod 4) or n == 2 (mod 4), then a(n) = 4n.

If p^2 divides n for some prime p, a(n) is a divisor of (4n)/p.

If n == 3 (mod 8), then a(n) is a divisor of n because numbers of the form x^2+n*y^2 cannot have any prime factors that are congruent to 2+n (mod 2n) raised to an odd power.

If n == 7 (mod 8), then a(n) is a divisor of 2n because numbers of the form x^2+n*y^2 can have prime factors that are congruent to 2+n (mod 2n) raised to an odd power, but they cannot be congruent to 2 (mod 4). So, we need to characterize the prime factor of 2 from the remaining prime factors that are congruent to 2+n (mod 2n) separately.

Examples

			For n = 7, consider the set of all residue classes to which a prime represented by the quadratic form x^2+7*y^2 belong to, {1, 9, 11, 15, 23, 25} mod 28. This can be simplified to {1, 9, 11} mod 14 and this is the lowest modulo this set of residue classes can be simplified to. So, a(7) = 14. x^2+7*y^2 is the only primitive quadratic form of discriminant = -28.
For n = 15, there are two quadratic forms of discriminant = -60, x^2+15*y^2 and 3*x^2+5*y^2. x^2+15*y^2 can be used to represent all primes in set of residue classes {1, 4} mod 15. 3*x^2+5*y^2 can be used to represent all primes in set of residue classes {3, 5, 17, 23} mod 30. The lowest common modulo is 30, because {1, 4} mod 15 can also be written as {1, 4, 16, 19} mod 30, and so a(15) = 30.
		

Crossrefs

Cf. A000926, A232550, A232551, A234002.