is composite for all natural numbers n. In 1960, Wacław Sierpiński proved that there are infinitely many odd integers k which have this property.
In other words, when k is a Sierpiński number, all members of the following set are composite: If the form is instead
Most currently known Sierpiński numbers possess similar covering sets.
[1] However, in 1995 A. S. Izotov showed that some fourth powers could be proved to be Sierpiński numbers without establishing a covering set for all values of n. His proof depends on the aurifeuillean factorization t4⋅24m+2 + 1 = (t2⋅22m+1 + t⋅2m+1 + 1)⋅(t2⋅22m+1 − t⋅2m+1 + 1).
In private correspondence with Paul Erdős, Selfridge conjectured that 78,557 was the smallest Sierpiński number.
[1] The distributed volunteer computing project PrimeGrid is attempting to eliminate all the remaining values of k:[5] In 1976, Nathan Mendelsohn determined that the second provable Sierpiński number is the prime k = 271129.
An ongoing search is trying to prove that 271129 is the second Sierpiński number, by testing all k values between 78557 and 271129, prime or not.