Talk:Strong prime

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Just in case anyone is wondering, 439351292910452432574786963588089477522344331 is in the ballpark of 439351292910452432574786963588089477522344721, the 138th prime of the Hoffman inversion of Wilf's primefree sequence. PrimeFan 23:12, 27 January 2006 (UTC)

The article says: "Given a twin prime {p, p + 2}, the lesser prime of the two, p, will almost certainly be a strong prime. In all the twin primes in the first ten million primes, the only lesser of a pair that is not a strong prime is 3 (from the pair {3, 5}, the arithmetic mean of 2 and 5 is 3.5)." In fact, {5, 7} is a pair of twin primes, and 5 is not strong according to the definition here (it's balanced, since 5 = (3+7)/2). In fact, 3 and 5 are the only lesser members of a pair of twin primes which are not strong primes. This is easy to show: if p and p+2 are primes, then p is not a strong prime if and only if one of p-1 or p-2 is prime. But say p >= 7. Then p is congruent to 1 or 5 mod 6. Since p+2 is prime, we must have p congruent to 5 mod 6. So neither p-1 nor p-2 is prime, since these are 4 and 3 mod 6, respectively. This proves that if p >= 7 is prime, and p+2 is prime, then p is strong. We can check p = 3, 5 "by hand", and we see that these aren't strong. I've edited the article accordingly. Izzycat 21:06, 16 February 2006 (UTC) ǖ


 * Thank you very much the correction and explanation. PrimeFan 22:02, 17 February 2006 (UTC)

Frequency of strong primes?
I think it would be very useful to give some guidance on the frequency / density of strong primes within ranges that are useful for cryptography (probably from hundreds of digits to a couple of thousand digits). Poking around with Google I have found some papers that appear to discuss this subject, but I am not qualified to interpret or summarize these papers. Perhaps someone else can update this article with this information? 98.234.112.116 (talk) 07:07, 16 May 2011 (UTC)

Strong prime definition in the Theory of numbers does not exist
Dears, are you sure that there is an universal and common definition of "Strong" prime within the Theory of Number? I'm much more confident that a definition is reported in some recent mathematical paper but it' s not common recognized by community. My suggestion is to keep as main definition what it's reported in the Cryttography section which explains better also why these numbers are called Strong. Later on for completeness it's possible to ention other small definition less important in this context and misleading for the idea you want to clarify here: why a prime number is called strong! . — Preceding unsigned comment added by Ctag67 (talk • contribs) 09:56, 11 October 2018 (UTC)


 * There is a definition of  (Number Theory)   strong primes   on the OEIS (The On-Line Encyclopedia of Integer Sequences®).
 * See the OEIS entry  strong   primes.


 * There is also a definition of  (Number Theory)   weak primes   on the OEIS (The On-Line Encyclopedia of Integer Sequences®).
 * See the OEIS entry  weak    primes.
 * -- GerardSchildberger (talk) 05:24, 3 December 2018 (UTC)

Weak primes are not defined
Weak primes  are directed to this article   (Strong primes),   but they are not defined herein. Perhaps someone should add a (number Theory) definition either here or add a weak primes entry. -- GerardSchildberger (talk) 05:31, 3 December 2018 (UTC)
 * Strong prime defines a weak prime. PrimeHunter (talk) 10:33, 3 December 2018 (UTC)

Split request
While I haven't added the tag, I certainly agree that the two different notions of a strong prime should be split to separate articles per WP:NOTDICT. However, sourcing for the notion from cryptography is already weak (one paper), and sourcing for the notion from number theory is completely absent. So we should have a bit more sourcing to demonstrate that both concepts are independently notable. Felix QW (talk) 20:48, 8 June 2024 (UTC)