TITLE:
A Proof of a Conjecture and Twenty-Five Conjectures in Number Theory
AUTHORS:
Zhongqi Zhou
KEYWORDS:
New Conjecture in Number Theory, A Generalization of Cullen’s Conjecture, Proof of the Conjecture, Computational Verification Methods
JOURNAL NAME:
Open Access Library Journal,
Vol.11 No.9,
September
27,
2024
ABSTRACT: 1) Fermat has proved that
x
4
+
y
4
=
z
2
has no positive integer solution, and in 2011, J. Cullen [1] reported that
x,y∈{
0,1,⋯,
10
7
}
,
x
4
+
y
4
+1
is not a square greater than 1, and conjecture:
x
4
+
y
4
+1≠
z
2
,
z∈{
2,3,⋯ }
,
x,y∈{
0,1,⋯ }
. On May 15, 2021, Sun Zhiwei [2] proposed that neither
x
4
+
y
4
+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.