Let us start with the observation that for m >= 2 the n-th centered 2m-gonal number is of the form (2m/2)*n*(n+1) + 1 (details here). Therefore, the centered 2m-gonal number C and the centered 2m-gonal number whose index is C, i.e. the number (2m/2)*C*(C+1) + 1, are coprime.
Therefore, any sequence of centered 2m-gonal numbers, whose first term a(1) is any centered 2m-gonal number and whose general term is of the form
a(n) = (2m/2)*C*(C+1) + 1, where C = Product_{i=1..n-1} a(i), is a sequence of pairwise coprime centered 2m-gonal numbers.
Corollary:
The above, by the Fundamental theorem of arithmetic, implies that there are infinitely many ways to prove the infinitude of primes.
Example:
Taking as a seed the centered square number 1 (see OEIS A001844), we can construct the following sequence of pairwise coprime centered square numbers:
1, 5, 61, 186661, 6482415409615261,
272402172694009346312913190157283525183169345861, ...
Therefore, any sequence of centered 2m-gonal numbers, whose first term a(1) is any centered 2m-gonal number and whose general term is of the form
a(n) = (2m/2)*C*(C+1) + 1, where C = Product_{i=1..n-1} a(i), is a sequence of pairwise coprime centered 2m-gonal numbers.
Corollary:
The above, by the Fundamental theorem of arithmetic, implies that there are infinitely many ways to prove the infinitude of primes.
Example:
Taking as a seed the centered square number 1 (see OEIS A001844), we can construct the following sequence of pairwise coprime centered square numbers:
1, 5, 61, 186661, 6482415409615261,
272402172694009346312913190157283525183169345861, ...
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