CF1423J Bubble Cup hypothesis

The Bubble Cup hypothesis stood unsolved for $ 130 $ years. Who ever proves the hypothesis will be regarded as one of the greatest mathematicians of our time! A famous mathematician Jerry Mao managed to reduce the hypothesis to this problem: Given a number $ m $ , how many polynomials $ P $ with coefficients in set $ {\{0,1,2,3,4,5,6,7\}} $ have: $ P(2)=m $ ? Help Jerry Mao solve the long standing problem!

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In first case, for $ m=2 $ , polynomials that satisfy the constraint are $ x $ and $ 2 $ . In second case, for $ m=4 $ , polynomials that satisfy the constraint are $ x^2 $ , $ x + 2 $ , $ 2x $ and $ 4 $ .