Twin prime conjecture

conjecture about the existence of infinitely many twin primes

The twin prime conjecture is a mathematical theory. It says that it is possible to find two twin primes that are as big as wanted.

Twin primes are prime numbers that differ by two. For example, 3 and 5 are both prime and differ by two. They are twin primes. 23 is prime, but it is not a twin prime. The primes nearest to 23 are 19 and 29. Twin primes were discovered by Euclid in 300 B.C.

Since Euclid's time mathematicians have wondered whether there are an infinite number of twin primes.[1] Many mathematicians are still trying to find the answer.

Here is Lawrence Abas' (Aurora Ontario Canada) proposed simple proof of infinite twin primes previously published on Linked-In with no current disputes.

This negative proof considers that a largest-twin-prime pair can only exist if there are a finite number of primes, which creates a paradox.

If there were a finite number of primes, the maximum twin prime would be easy to calculate the result of multiplying that set of primes (I call it a prime factorial) plus one or minus one. This this would produce two new numbers that cannot be in that set or have factors that are in that set. Since this makes new prime numbers that are not in the set or composite numbers that have factors that are not in that set a paradox exists.

Consider a finite set a set of unique sequential prime numbers where each element is multiplied together. The result minus one, and result plus one cannot be an element in that set. Example: (2,3,5, 7) 2 x 3 x 5 x 7 = 210 210-1=209, 210+1=211. 209 is 11x19 and 211 is prime. 11, 19, and 211 are primes that are not in the set (2,3,5,7). This creates a contradiction as in Euclid's proof of infinite primes, that any set of prime numbers can be used to find more prime numbers. When sequential primes are used this ensure the factors found are greater any prime in the set.

Note that the set of numbers described by n x 210 +/- 1 where n is a natural number from 1 to infinity, are twin prime candidates and guarantees that is cannot have the factors 2,3,5, and 7, not that any pair is a twin prime.

Since we know from Euclid that an infinite number of primes exist, the maximum twin primes is on n x (the result of all prime numbers multiplied, which is infinity) plus one and minus one. Where n is a natural number from 1 to infinity.

Any natural number multiplied by infinity is infinity, and infinity plus or minus one is still infinity. Therefore, the largest twin prime is infinite.

This negative proof is essentially identical to Euclid's proof of infinite primes except that +1 is changed to +1 and -1 below:

Consider any finite list of prime numbers p1, p2, ..., pn. I

t will be shown that at least one additional prime number not in this list exists.

Let P be the product of all the prime numbers in the list: P = Let q = P + 1 and Let r = P – 1:

Then q or r is either prime or not:

·      If q or r is prime, then there is at least one more prime that is not in the list, namely, q itself.

·      If q or r is not prime, then some prime factor p divides q. If this factor p were in our list, then it would divide P (since P is the product of every number in the list); but p also divides P + 1 = q, as just stated. If p divides P and also q, then p must also divide the difference of the two numbers, which is (P + 1) − P or just 1. Since no prime number divides 1, p cannot be in the list. This means that at least one more prime number exists beyond those in the list.

·      If q and r are prime then there then there is at least two more prime that is not in the list, namely, q  and  r itself.

This proves that for every finite list of prime numbers there is a prime number not in the list.

Quite simply, a maximum twin prime cannot exist with an infinite set of prime numbers.


  1. McKee, Maggie (14 May 2013). "First proof that infinitely many prime numbers come in pairs". Nature: International Weekly Journal of Science. doi:10.1038/nature.2013.12989. S2CID 124113283. Retrieved 9 July 2018.