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ab + (a + 1) (b + 1) = 2^k diophantine and k+1 prime

Source: INAMO Shortlist 2014 N1

July 9, 2019
number theoryDiophantine equationprime

Problem Statement

(a) Let kk be an natural number so that the equation ab+(a+1)(b+1)=2kab + (a + 1) (b + 1) = 2^k does not have a positive integer solution (a,b)(a, b). Show that k+1k + 1 is a prime number. (b) Show that there are natural numbers kk so that k+1k + 1 is prime numbers and equation ab+(a+1)(b+1)=2kab + (a + 1) (b + 1) = 2^k has a positive integer solution (a,b)(a, b).
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