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2009 Silk Road
4
4
Part of
2009 Silk Road
Problems
(1)
(x^2+p t^2)(y^2+p t^2)(z^2+p t^2) is a perfect square when (x, y, z, t) distinct
Source: SRMC 2009
9/2/2018
Prove that for any prime number
p
p
p
there are infinitely many fours
(
x
,
y
,
z
,
t
)
(x, y, z, t)
(
x
,
y
,
z
,
t
)
pairwise distinct natural numbers such that the number
(
x
2
+
p
t
2
)
(
y
2
+
p
t
2
)
(
z
2
+
p
t
2
)
(x^2+p t^2)(y^2+p t^2)(z^2+p t^2)
(
x
2
+
p
t
2
)
(
y
2
+
p
t
2
)
(
z
2
+
p
t
2
)
is a perfect square.
number theory
Perfect Square
prime