Properties of a perfect sfourth

Number Theory Level pending

$a_1^4=a_2^4+a_3^4- (a_4^2+a_5^2)$ is this true for all: $a_1≥3$ where all variables are not necessarily distinct positive integers. note it is asking if a perfect fourth≥81 can be expressed as the sum of two perfect fourths plus or minus sum of two squares

no yes

1 solution

Aareyan Manzoor
Nov 14, 2015

first, claim: $a^2=b^2-c^2$ for all a≥3. why, $a^2=(b-c)(b+c)$ here, when a is odd, we can just put b-c=1 and b+c= $a^2$ to get integer solutions. in the case a is even, we see that $4|a^2$ . so if we put b-c=2, then b+c will be even, giving us integer solution. this works from a=3 as 2 and 1 yields either b or c to be 0.. so we have proved this. now back to the original eqn, square both sides $a^4=b^4+c^4-2b^2c^2=b^4+c^4-(b^2c^2+b^2c^2)$ this is the same thing as the question so.. yes!

Properties of a perfect sfourth

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