A Proof of a Conjecture and Twenty-Five Conjectures in Number Theory

1) Fermat has proved that x⁴ y⁴=z²   has no positive integer solution, and in 2011, J. Cullen [1] reported that x,y,∈{0,1,...,10⁷}, x⁴ y⁴ 1 is not a square greater than 1, and conjecture:x⁴ y⁴ 1≠z²,z∈{2,3,...},x,y,∈{0,1,...}. On May 15, 2021, Sun Zhiwei [2] proposed that neither x⁴ y⁴ 1(x,y,∈N) is a perfect power based on Cullen’s conjecture (the form is z^m,(z,m∈{2,3...}) called perfect power). This paper generalizes and proves J. Cullen’s conjecture. 2) A lot of data calculation and verification are carried out, and 25 conjectures in number theory are put forward for number theory lovers to study.