已知a,b,c,d均为正整数,且a^5=b^4,c^3=d^2,a-c=65,求b-d

Question

已知a,b,c,d均为正整数,且a^5=b^4,c^3=d^2,a-c=65,求b-d

Answer 1

设m^4=a m^5=b
n^2=c n^3=d (m,n均为正整数)
则m^4-n^2=65
(m^2+n)(m^2-n)=65
易知m^2+n=13 m^2-n=5 (分解质因数,如果m^2+n=65,m^2-n=1,则m非整数)
解得m=3 n=4
所以b=m^5=243 d=n^3=64
b-d=179