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 ways of proving 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 ways of proving 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, ...}.
RSS Feed
