Editing Openai/68ec50da-cf00-8005-b5f6-b683506e5853 (section)

==== Wolfram|Alpha sometimes handles “over primes” products directly. Try the natural‑language versions first: ====
* Natural‑language query (in one line): ``<code> product over primes p with p ≡ 1 mod 4 and p > 17 of (1 - 2/p^2) </code>`<code> If it struggles, ask for a truncated version: </code>`<code> product over primes p with p ≡ 1 mod 4 and 17 < p ≤ 10^6 of (1 - 2/p^2) numeric approximation <syntaxhighlight>- Mathematica‑style input that WA usually accepts (finite truncation): </syntaxhighlight> N[Times @@ (1 - 2/#^2 & /@ Select[Prime[Range[2000000]], Mod[#,4]==1 && #>17 &]), 20] </code>`<code> Explanation: Prime[Range[2000000]] lists the first 2,000,000 primes; Select[... , Mod[#,4]==1 && #>17 &] filters to p≡1(mod4)p\equiv1 \pmod 4p≡1(mod4) and p>17p>17p>17; Times @@ (1 - 2/#^2 & /@ ...) multiplies (1−2/p2)(1-2/p^2)(1−2/p2) over that list; N[...,20] asks for 20 digits. Increase 2000000 until the value stabilizes to the digits you want.
* Pure natural‑language, asking for a limit via truncation: </code>`<code> limit as x→∞ of product over primes p with p ≡ 1 mod 4 and 17 < p ≤ x of (1 - 2/p^2) </code>``

If WA refuses the infinite product directly, use the truncated version with a large upper limit (e.g. 10610^6106 or 10710^7107); by the bound above the tail error is at most about 2/B2/B2/B, so with B=106B=10^6B=106 your error is ≲2×10−6\lesssim 2\times 10^{-6}≲2×10−6.

===== - With cutoff B=106B=10^6B=106: product ∈[0.991159382557751, 0.9911613648784985]\in [0.991159382557751,\ 0.9911613648784985]∈[0.991159382557751, 0.9911613648784985]. =====
* With cutoff B=107B=10^7B=107: product ∈[0.991161105238931, 0.9911613034711718]\in [0.991161105238931,\ 0.9911613034711718]∈[0.991161105238931, 0.9911613034711718].

These are rigorous intervals for the infinite product.