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In the spirit of "examples do not prove anything in full generality (even if you have billions of them)", here is my favorite one: $gcd(n^{17}+9,(n+1)^{17}+9)=1$ is true for all integers $1\leq n<8424432925592889329288 197322308900672459420460792433$, but false for $n=8424432925592889329288 197322308900672459420460792433$.