Different proofs that show $n ( n + 1 ) ( n + 2 ) ( n + 3 )$ cannot be the square of an integer, where n is a natural number.
2026-04-01 11:32:50.1775043170
Different proofs for $n ( n + 1 ) ( n + 2 ) ( n + 3 )$
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$$n(n+1)(n+2)(n+3)=n(n+3)(n+1)(n+2)=(n^2+3n)(n^2+3n+2)=$$ Let $n^2+3n=a$ $$=a(a+2)=a^2+2a$$ $$a^2<a^2+2a<a^2+2a+1=(a+1)^2$$ $$(n^2+3n)^2<n(n+1)(n+2)(n+3)<(n^2+3n+1)^2$$