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Twin Primes Finder

Enter an upper limit to list every twin prime pair below it.

How it is calculated

The tool runs a Sieve of Eratosthenes up to your limit, marking every multiple of each prime as composite. It then scans the primes p and checks whether p + 2 is also prime, counting a pair when both are at or below the limit. It reports the total count, the largest pair and the first 50 pairs, with an ellipsis if there are more. The limit must be at least 5, is cut down to a whole number, and anything above 200,000 is silently treated as 200,000. The pair (3, 5) is included, and so is (5, 7), since the tool counts every pair of primes that differ by 2, even when they overlap.

Worked example

With a limit of 100 the sieve leaves the primes up to 100. Scanning for gaps of 2 gives (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61) and (71, 73). That makes 8 twin prime pairs, and the largest is (71, 73). The next pair, (101, 103), lies just beyond the limit so it is not counted.

Try this example →

Common mistakes

  • Entering a limit below 5 and expecting a result; there is no pair to find, so the tool shows nothing.
  • Expecting a limit above 200,000 to be honoured. Larger values are capped at 200,000, so the count and largest pair describe that range.
  • Counting a pair when only its lower prime fits. Both numbers must be at or below the limit, so a limit of 72 excludes (71, 73) and gives 7 pairs, while 73 gives 8
  • Looking for pairs with other gaps, such as 4 (cousin primes) or 6; only a difference of exactly 2 qualifies.
  • Reading the pair list as complete when it ends with an ellipsis. Only the first 50 pairs are shown, though the count covers all of them.

About this calculator

Twin primes are pairs of prime numbers that differ by exactly 2, such as (3,5), (11,13), and (17,19). Finding them up to a limit N is typically done by first sieving all primes below N (the Sieve of Eratosthenes) and then scanning the resulting list for consecutive primes p and p+2 — far faster than trial-dividing every candidate individually.

This finder takes an upper limit, runs the sieve, lists every twin prime pair below it, and reports the total count. Students and hobbyists studying number theory use it to explore the (still unproven) twin prime conjecture — that infinitely many such pairs exist — and to build intuition about how prime density thins out as numbers grow, a question with deep ties to sieve theory and, more distantly, to the difficulty assumptions behind prime-based cryptography.

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  • Editorial team

    Try a few limits to watch the pairs thin out: below 100 there are 8 pairs, below 1,000 there are 35 and below 10,000 there are 205. Because (3, 5) and (5, 7) share the prime 5, the count is of pairs, not of distinct primes. The tool stops at 200,000, so a larger limit returns the same answer as 200,000. Do you have a pattern you want to check, such as pairs around a specific number?

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