# Concept derivation guides

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Concept guide



## [Mathematical proof: proving n²+n is always even, and refuting 'n²+n+41 is always prime' by a counterexample](/en/derivations/derivation-mko-proof-01/?from=%2Fen%2Fderivations%2F%3Fq%3Dmko-proof)

(1) In a stated context for the integers, prove from explicit premises and rules the theorem T: for every integer n, 2∣(n²+n); then derive from T by contradiction that no integer n makes n²+n odd. (2) Test the claim C: 'for every natural number n, n²+n+41 is prime'; explain why the 40 positive cases n=0 to 39 do not constitute a proof, and refute C by the counterexample n=40.

Mathematical proof

derivation-mko-proof-01





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