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Number Theory Problem Set I
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Number Theory Problem Set I
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Number Theory Problem Set IMaths Olympiad Preparation
17th February 2024
1 Problems
1. Prove that there are infinitely many positive integers (a, b, c) in
arithmetic progression such that ab + 1, bc + 1 & ca + 1 are all per-
fect squares.
2. We define a series an as follows
a0 = −1, a1 = 0, a2 = 12, an+3 + 55an+1 = 14an+2 + 42an .
Prove that 43 divides ap , where p is a prime greater than 3.
3. What is the least positive integer n such that 25n + 16n leaves a
remainder of 1 when divided by 121?
4. Find all positive integers x, y so that (x2 + y)(y 2 + x) is the 5th
power of a prime number.
5. Solve in positive integers the equation
1 1 2
mn + nm = 2 + 1 1 .
mn(m + n) m + n
6. A sequence of positive reals {an } is defined below
a2n+1 + 2
a0 = 1, a1 = 3, an+2 = .
an
Show that for all nonnegative integer n, an is a positive integer.
Happy Problem Solving!
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2 Solutions
1. Let (p, q) be a non-fundamental solution of the pell’s equation
x2 − 3y 2 = 1.
Let (a, b, c) = (2q − p, 2q, 2q + p).
Claim I: a is positive.
Proof: We need to prove that
2q > p,
i.e. 4q 2 > p2 = 3q 2 + 1,
i.e. q 2 > 1.
Which is clearly true.
Claim II: ab + 1, bc + 1 & ca + 1 are all perfect squares.
Proof: We have,
ab + 1 = (2q − p)2q + 1
= 4q 2 − 2pq + 1
= 4q 2 − 2pq + p2 − 3q 2
= (p − q)2 .
bc + 1 = 2q(2q + p) + 1
= 4q 2 + 2pq + 1
= 4q 2 + 2pq + p2 − 3q 2
= (p + q)2 .
ca + 1 = (2q + p)(2q − p) + 1
= 4q 2 − p2 + 1
= 4q 2 − p2 + p2 − 3q 2
= q2 .
Hence, the desired result follows.
QED.
2 Maths Olympiad Preparation
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