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Sierpiński number - mklabgr - 07-10-2026
Sierpiński number
Summary
A Sierpiński number is an odd positive integer (k) with the remarkable property that every number of the form ($k \times 2^n + 1$) is composite, no matter which natural number (n) is chosen. Introduced through the work of Polish mathematician Wacław Sierpiński in 1960, these numbers occupy an important place in number theory because they demonstrate that certain infinite families of integers can never produce prime numbers.
Many known Sierpiński numbers can be verified using a finite collection of prime divisors, called a covering set, which guarantees that every value in the sequence has at least one nontrivial factor. Some special cases, however, require more advanced techniques, such as Aurifeuillean factorization, instead of a traditional covering set.
One of the most intriguing open questions is the Sierpiński problem, which asks for the smallest Sierpiński number. The leading candidate, $78,557$, was identified by John Selfridge in 1962 and is widely believed to be the smallest example, although a complete proof still depends on showing that several smaller candidates eventually generate a prime of the form ($k \times 2^n + 1$).
Distributed computing projects such as PrimeGrid continue this search, while related challenges investigate the smallest prime Sierpiński number and numbers that are simultaneously Sierpiński and Riesel numbers. These ongoing investigations highlight how seemingly simple numerical patterns can inspire deep mathematical research, large-scale computational collaboration, and a richer understanding of prime numbers and their distribution.
ARTICLE