perplexus.info :: Just Math : Confirm or negate

Confirm or negate (Posted on 2020-04-24)

There is no polynomial P(x) with integer coefficients such that P(7)=11 and P(11)=13.

Prove or provide a counterexample.

No Solution Yet Submitted by Ady TZIDON

Rating: 5.0000 (1 votes)

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solution

| Comment 1 of 11

Plug in x=11 and x=7 and subtract the two equations.

Every term of the difference is integer*(11^k - 7^k) which is divisible by 11-7=4.

But the difference is 13-11=2 which is not divisible by 4, so no such polynomial exists.

Posted by xdog on 2020-04-24 16:39:35

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