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Miklós Schweitzer
2005 Miklós Schweitzer
3
3
Part of
2005 Miklós Schweitzer
Problems
(1)
diophantine eqn
Source: Miklos Schweitzer 2005 q3
8/10/2021
Let
α
≤
22
\alpha\leq 22
α
≤
22
be non-negative integer. For which
α
\alpha
α
does the equation
8
x
23
−
5
α
y
23
=
1
8x^{23}-5^{\alpha}y^{23}=1
8
x
23
−
5
α
y
23
=
1
have the most integer solutions (x,y)? What can we say about
α
≥
23
\alpha\geq 23
α
≥
23
?I believe the eqn has solutions only when
α
=
0
\alpha=0
α
=
0
. taking modulo 47,
α
≡
9
,
17
(
m
o
d
23
)
\alpha\equiv 9,17\pmod{23}
α
≡
9
,
17
(
mod
23
)
or (
23
∣
α
23|\alpha
23∣
α
and
47
∣
x
47|x
47∣
x
). taking modulo 139 and 277 eliminates the
α
≡
9
,
17
(
m
o
d
23
)
\alpha\equiv 9,17\pmod{23}
α
≡
9
,
17
(
mod
23
)
cases. 139=23*6+1 , 277=23*12+1
number theory