Number of Solutions

Number Theory Level 3

Find the number of pairs of odd primes p , q p, q such that

p q − q p = p + q p^q-q^p=p+q

8 0 4 2

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Alan Yan by Alan Yan , 20, USA

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1 solution

Alan Yan
Oct 10, 2015

Take modulo p p and by Fermat's Little Theorem, you get that q ≡ − q ⟹ p = q q \equiv -q \implies p = q but this is obviously wrong. Therefore there are no solutions.

2 Helpful 0 Interesting 0 Brilliant 1 Confused

What about 2 5 − 5 2 = 2 + 5 2^5-5^2=2+5 ? I somehow got this question wrong with the answer 0 even though I don't remember chosing it.

Xuming Liang - 5 years, 8 months ago

modulo p gives p=q, if p is not 2. If p=2,then p=2 & q=5 satisfy the given equation.So how can you say there are no solutions? One pair (2,5) is the correct answer.

vinod trivedi - 5 years, 8 months ago

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Oops, meant to add odd primes.

Alan Yan - 5 years, 8 months ago

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