Twin prime numbers are two prime numbers that differ by exactly 2. For example, 11 and 13 are twin primes because both are prime and 13 minus 11 is 2. Other pairs are (3, 5), (5, 7), (17, 19) and (29, 31).
The chart below shows twin prime pairs. Each pair sits as close together as two odd primes can.
- 1(3, 5)
- 2(5, 7)
- 3(11, 13)
- 4(17, 19)
- 5(29, 31)
- 6(41, 43)
- 7(59, 61)
- 8(71, 73)
- 9(101, 103)
- 10(107, 109)
- 11(137, 139)
- 12(149, 151)
- 13(179, 181)
- 14(191, 193)
- 15(197, 199)
- 16(227, 229)
- 17(239, 241)
- 18(269, 271)
- 19(281, 283)
- 20(311, 313)
- 21(347, 349)
- 22(419, 421)
- 23(431, 433)
- 24(461, 463)
- 25(521, 523)
- 26(569, 571)
- 27(599, 601)
- 28(617, 619)
- 29(641, 643)
- 30(659, 661)
- 31(809, 811)
- 32(821, 823)
- 33(827, 829)
- 34(857, 859)
- 35(881, 883)
What are twin prime numbers?
To be twin primes, a pair must pass two tests:
- Both numbers must be prime numbers.
- The difference between them must be exactly 2.
Why 2? Apart from 2 itself, every prime is odd. Two odd numbers can never be just 1 apart, so 2 is the smallest possible gap between two odd primes. The pair (2, 3) differs by 1, but it is not counted as twin primes.
Some pairs fail the test. 7 and 9 differ by 2, but 9 = 3 × 3 is not prime. 23 and 25 also fail because 25 = 5 × 5.
Twin prime numbers from 1 to 100
There are 8 twin prime pairs from 1 to 100:
(3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71, 73)
- From 1 to 50 there are 6 pairs: (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43).
- From 51 to 100 there are 2 pairs: (59, 61), (71, 73).
Notice that 5 belongs to two pairs, (3, 5) and (5, 7). It is the only prime that is part of two twin pairs. The numbers 3, 5 and 7 form the only prime triplet where each step is 2, because in any three odd numbers in a row like this, one of them always divides by 3.
Twin primes from 101 to 200 and beyond
Between 101 and 200 there are 7 more pairs: (101, 103), (107, 109), (137, 139), (149, 151), (179, 181), (191, 193), (197, 199). So there are 15 twin prime pairs below 200. Up to 1000 there are 35 pairs in all, and the last one below 1000 is (881, 883).
Mathematicians believe twin primes go on forever, but nobody has proved it yet. This open question is called the twin prime conjecture. It has been studied for a very long time, and recent work has shown that there are endless pairs of primes with some fixed gap, but the gap of 2 is still unproved.
A handy pattern
Every twin prime pair after (3, 5) has the form (6k minus 1, 6k plus 1). In other words, the number between the two twins is always a multiple of 6. Check it:
| Twin pair | Number between | Multiple of 6 |
|---|---|---|
| (5, 7) | 6 | 6 × 1 |
| (11, 13) | 12 | 6 × 2 |
| (17, 19) | 18 | 6 × 3 |
| (29, 31) | 30 | 6 × 5 |
| (41, 43) | 42 | 6 × 7 |
| (59, 61) | 60 | 6 × 10 |
| (71, 73) | 72 | 6 × 12 |
This does not mean every multiple of 6 sits between twins. For 24, the neighbours are 23 and 25, and 25 is not prime. But it does make twin primes quicker to hunt for: look either side of 6, 12, 18, 24 and so on, and test both numbers.
How to find twin primes
- Write down the prime numbers 1 to 100 (or any range).
- Look for two primes next to each other in the list with a gap of 2.
- Write each such pair in brackets.
Example: in the primes 41, 43, 47, the pair (41, 43) has a gap of 2, so it is a twin pair. The gap from 43 to 47 is 4, so (43, 47) is not.
Twin primes and co-primes
Students often mix up twin primes and co-prime numbers. Twin primes must both be prime and differ by 2. Co-primes only need to share no common factor except 1, and they do not have to be prime at all. Every twin prime pair is also a co-prime pair, because two different primes never share a factor. But 8 and 9 are co-prime without being twin primes.
Test any number with the prime checker below, then try its neighbours 2 above and 2 below.
Questions people ask
What are twin prime numbers?
Twin prime numbers are two prime numbers whose difference is exactly 2. Examples are (3, 5), (11, 13) and (41, 43). Both numbers in the pair must be prime.
What is the definition of twin prime numbers?
A pair of primes p and p + 2 is called a twin prime pair. The gap of 2 is the smallest possible gap between two odd primes. The first pair is (3, 5).
How many twin prime pairs are there from 1 to 100?
There are 8 twin prime pairs from 1 to 100: (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71, 73).
How many twin prime numbers are there from 1 to 50?
There are 6 twin prime pairs from 1 to 50: (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43).
Are (2, 3) twin primes?
No. 2 and 3 are both prime, but their difference is 1, not 2. Twin primes must differ by exactly 2, so the first pair is (3, 5).