Elaborating on my previous text called "One easy way to prove the infinitude of primes", I found the following
Theorem
Let m = 2*l, for any l > 0. There are infinitely many sequences of pairwise coprime m-gonal numbers, whose first term a(1) is any positive m-gonal number and whose general term a(n) is of the form a(n) = (k +1)*((l – 1)*k + 1)), where
k = Product_{i=1..n-1} a(i).
Corollary
The fact that any such sequence is infinite implies (by the Fundamental theorem of arithmetic) the infinitude of primes.
Example
Let us take for example the Tetradecagonal numbers (where m = 14 and l = 7) and take the second 14-gonal number 14 as the first term of the new sequence (NS). The general term of NS is of the form a(n) = (k +1)*(6k + 1),
where k = Product_{i=1..n-1} a(i) and
NS = {14, 645, 244658821, 14642610579551886703145221, ...}.
Theorem
Let m = 2*l, for any l > 0. There are infinitely many sequences of pairwise coprime m-gonal numbers, whose first term a(1) is any positive m-gonal number and whose general term a(n) is of the form a(n) = (k +1)*((l – 1)*k + 1)), where
k = Product_{i=1..n-1} a(i).
Corollary
The fact that any such sequence is infinite implies (by the Fundamental theorem of arithmetic) the infinitude of primes.
Example
Let us take for example the Tetradecagonal numbers (where m = 14 and l = 7) and take the second 14-gonal number 14 as the first term of the new sequence (NS). The general term of NS is of the form a(n) = (k +1)*(6k + 1),
where k = Product_{i=1..n-1} a(i) and
NS = {14, 645, 244658821, 14642610579551886703145221, ...}.
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