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Nepal TST
2024 Nepal TST
P1
P1
Part of
2024 Nepal TST
Problems
(1)
Exponents modulo 4
Source: 2024 Nepal TST P1
4/12/2024
Let
a
,
b
a, b
a
,
b
be positive integers. Prove that if
a
b
+
b
a
≡
3
(
m
o
d
4
)
a^b + b^a \equiv 3 \pmod{4}
a
b
+
b
a
≡
3
(
mod
4
)
, then either
a
+
1
a+1
a
+
1
or
b
+
1
b+1
b
+
1
can't be written as the sum of two integer squares.(Proposed by Orestis Lignos, Greece)
number theory
Parity
modular arithmetic