Euclid-Mullin sequence

Euclid-Mullin sequence

The Euclid-Mullin sequence is an infinite sequence of distinct prime numbers, in which each element is the least prime factor of one plus the product of all earlier elements.

The first 43 elements of the sequence are OEIS|id=A000945:

:2, 3, 7, 43, 13, 53, 5, 6221671, 38709183810571, 139, 2801, 11, 17, 5471, 52662739, 23003, 30693651606209, 37, 1741, 1313797957, 887, 71, 7127, 109, 23, 97, 159227, 643679794963466223081509857, 103, 1079990819, 9539, 3143065813, 29, 3847, 89, 19, 577, 223, 139703, 457, 9649, 61, 4357

These are the only known elements as of 2008. Finding the next one requires finding the least prime factor of a 180-digit number (which is known to be composite).

Definition

If "an" denotes the "n"-th element of the sequence, then "an" is the least prime factor of:left(prod_{i < n} a_i ight)+1,.The first element is therefore the least prime factor of the empty product plus one, which is 2. The element 13 in the sequence is the least prime factor of 2 × 3 × 7 × 43 + 1 = 1806 + 1 = 1807 = 13 × 139.

Properties

The sequence is infinitely long and does not contain repeated elements. This can be proved using the method of Euclid's proof that there are infinitely many primes. In fact, that proof is constructive, and the sequence is the result of performing a version of that construction.

Conjecture

It may be conjectured that every prime number appears in the Euclid-Mullin sequence. However, there is currently no insight into how a proof of the conjecture might be approached. The least prime number not known to be an element of the sequence is 31.

The positions of the prime numbers from 2 to 97 are::1, 2, 7, 3, 12, 5, 13, 36, 25, 33, ?, 18, ?, 4, ?, 6, ?, 42, ?, 22, ?, ?, ?, 35, 26 (OEIS2C|id=A056756)where ? indicates that the position (or whether it exists at all) is unknown as of 2008. (The listing with the question marks is given in the Extensions field, whereas the main listing stops at 33 and has no question marks).

ee also

*Euclid number
*Sylvester's sequence

References

*MathWorld|title=Euclid-Mullin Sequence|urlname=Euclid-MullinSequence

External links

* [http://mersenneforum.org/showpost.php?p=60960&postcount=65 "Factoring 43rd Term of Euclid-Mullin sequence"] , factorizations of the numbers for which the sequence elements is the smallest prime factor


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