A373240 Greatest common divisor of the product of terms in positive Pythagorean n-tuples, over all possible such tuples.

60, 12, 2, 24, 6, 8, 12, 3, 4, 12, 6, 8, 6, 24, 4, 3, 12, 4, 6, 24, 2, 24, 12, 1

Offset: 3

Author

Brian Almond, May 28 2024

Keywords

• nonn
• more

Comments

A Pythagorean n-tuple consists of positive integers X1,...,Xn where X1^2 + X2^2 + ... + X(n-1)^2 = Xn^2. The product of those terms is P = X1*X2*...*Xn.

a(n) is the GCD of all possible products P arising this way.

If n is odd, then 2 | a(n).

If n mod 3 is 0 or 1, then 3 | a(n).

If n is divisible by 4, then 4 | a(n).

If n mod 8 is 1 or 3, then 4 | a(n).

If n mod 8 is 0 or 6, then 8 | a(n).

If n mod 24 is 2, then a(n)=1.

a(n+24)=a(n) for n >= 4.

Examples

			For n = 8, then 8 | a(n).  Since 1^2 + 1^2 + 1^2 + 1^2 + 2^2 + 2^2 + 2^2 = 4^2 with P = 1*1*1*1*2*2*2*4 = 32 and 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 2^2 + 4^2 = 5^2 with P = 1*1*1*1*1*2*4*5 = 40, then a(8) = gcd(32,40) = 8, and no larger number will divide the product of terms in every Pythagorean octuple.
		

Links

• Des MacHale and Christian van den Bosch, Generalising a result about Pythagorean triples, The Mathematical Gazette, Vol. 96, March 2012.