Little math problems September 2018

[Albert Chan]

(09-25-2018 12:48 AM)Valentin Albillo Wrote:         3430000 ^2 + 10290000 ^2 = 117649000000000 = 49000 ^3
      3430000 ^3 + 10290000 ^3 = 1129900996000000000000 = 33614000000 ^2

This is how above solution (a, 3 a) is obtained:

3² + 1 = 10 = 2 * 5
3³ + 1 = 28 = 2² * 7

So, a may be the form 2^x 5^y 7^z
Sum of square = 2^2x+1 5^2y+1 7^2z
Sum of square is cube, thus x=1, y=1, z=0 (modulus 3)

Sum of cubes = 2^3x+2 5^3y 7^3z+1
Sum of cubes is square, thus x=0, y=0, z=1 (modulus 2)

Minimum (x, y, z) that satisfy both => (4, 4, 3), thus a = 2⁴ 5⁴ 7³ = 3430000