A268511 Odd integers n such that 3^n + 5^n = x^2 + y^2 (x and y integers) is solvable.

1, 5, 13, 17, 29, 89, 109, 149, 157, 193, 373

Offset: 1

Author

Altug Alkan, Feb 06 2016

Keywords

• nonn
• more

Comments

Corresponding 3^n + 5^n values are 8, 3368, 1222297448, 763068593288, 186264583553473068008, ...

445 <= a(12) <= 509. 509, 661, 709 are terms. - Chai Wah Wu, Jul 22 2020

Examples

			1 is a term because 3^1 + 5^1 = 8 = 2^2 + 2^2.
5 is a term because 3^5 + 5^5 = 3368 = 2^2 + 58^2.
13 is a term because 3^13 + 5^13 = 1222297448 = 4118^2 + 34718^2.
		

Crossrefs

Cf. A001481, A074606.

Programs

Mathematica

Select[Range[1, 110, 2], Resolve@ Exists[{x, y}, Reduce[3^# + 5^# == (x^2 + y^2), {x, y}, Integers]] &] (* Michael De Vlieger, Feb 07 2016 *)

PARI

is(n) = #bnfisintnorm(bnfinit(z^2+1), n);
for(n=1, 1e3, if(n%2==1 && is(3^n + 5^n), print1(n, ", ")));

Python

from sympy import factorint
A268511_list = []
for n in range(1,50,2):
    m = factorint(3**n+5**n)
    for d in m:
        if d % 4 == 3 and m[d] % 2:
            break
    else:
        A268511_list.append(n) # Chai Wah Wu, Dec 26 2018

Extensions

a(8)-a(9) from Giovanni Resta, Apr 10 2016

a(10)-a(11) from Chai Wah Wu, Jul 22 2020