Puzzle 1268 Primes and e

Problems & Puzzles: Puzzles

Puzzle 1268 Primes and e

On March 16, 2026, Sebastián Martín Ruiz wrote:

Let p(n) the nth prime number

Q1) Prove that Limit n->Infinity (p(n)#)^(1/p(n))=e

* p(n)#=p(1)·p(2)·p(3)·...·p(n) (primorial)

Q2) Prove that Limit n->Infinity Product(ci,ci composite<=n)^(1/p(n))=e

			From May 17 to 23, 2026, contributions came from Emmanuel Vantieghem

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Emmanuel wrote:

				To prove the "theorems" in Puzzle 1268 I had to dig in my old 
				textbooks on Analytic Number Theory (which I read out of pure 
				curiosity since my "domain" was that of Algebraic Number 
				Theory).

				I only hope that Sebastian will not be disappointed that I did 
				not include the proofs of the "well known results" I used 
				further on.

				Here is what I found :




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