Sierpinski number

Known Sierpiński numbers

The sequence of currently known Sierpiński numbers begins with:

78557, 271129, 271577, 322523, 327739, 482719, 575041, 603713, 903983, 934909, 965431, 1259779, 1290677, 1518781, 1624097, 1639459, 1777613, 2131043, 2131099, 2191531, 2510177, 2541601, 2576089, 2931767, 2931991, ... (sequence A076336 in the OEIS).

The number 78557 was proved to be a Sierpiński number by John Selfridge in 1962, who showed that all numbers of the form 78557⋅2ⁿ + 1 have a factor in the covering set {3, 5, 7, 13, 19, 37, 73}. For another known Sierpiński number, 271129, the covering set is {3, 5, 7, 13, 17, 241}. All currently known Sierpiński numbers possess similar covering sets.

Sierpiński problem

The Sierpiński problem is: "What is the smallest Sierpiński number?"

In 1967, Sierpiński and Selfridge conjectured that 78,557 is the smallest Sierpiński number, and thus the answer to the Sierpiński problem.

To show that 78,557 really is the smallest Sierpiński number, one must show that all the odd numbers smaller than 78,557 are not Sierpiński numbers. That is, for every odd k below 78,557 there exists a positive integer n such that k2ⁿ+1 is prime. As of November 2016, there are only five candidates:

k = 21181, 22699, 24737, 55459, and 67607

which have not been eliminated as possible Sierpiński numbers. Seventeen or Bust (with PrimeGrid), a distributed computing project, is testing these remaining numbers. If the project finds a prime of the form k2ⁿ + 1 for every remaining k, the Sierpiński problem will be solved.

Since the second proved Sierpiński number is 271129, there is also a second Sierpiński number search project. The unknown values of k between 78557 and 271129 are

79309, 79817, 91549, 99739, 131179, 152267, 156511, 163187, 168451, 193997, 200749, 202705, 209611, 222113, 225931, 227723, 229673, 237019, 238411

Smallest n for which k×2n+1 is prime

0, 0, 1, 0, 1, 0, 2, 1, 1, 0, 1, 0, 2, 1, 1, 0, 3, 0, 6, 1, 1, 0, 1, 2, 2, 1, 2, 0, 1, 0, 8, 3, 1, 2, 1, 0, 2, 5, 1, 0, 1, 0, 2, 1, 2, 0, 583, 1, 2, 1, 1, 0, 1, 1, 4, 1, 2, 0, 5, 0, 4, 7, 1, 2, 1, 0, 2, 1, 1, 0, 3, 0, 2, 1, 1, ... (sequence A040076 in the OEIS) or  A078680 (not allow n = 0), for odd ks, see  A046067 or  A033809 (not allow n = 0).

For more terms k ≤ 1200, see [1] (k ≤ 300), [2] (301 ≤ k ≤ 600), [3] (601 ≤ k ≤ 900), and [4] (901 ≤ k ≤ 1200).

Simultaneously Sierpiński and Riesel

A number may be simultaneously Sierpiński and Riesel. The smallest five known examples are 3316923598096294713661, 10439679896374780276373, 11615103277955704975673, 12607110588854501953787, 17855036657007596110949, ... (A076335).

Dual Sierpinski problem

If we take the n of k2ⁿ + 1 to a negative integer, then the number become 2 − n + k 2 − n . If we choose the numerator, then the number become 2⁻ⁿ + k. Thus, a dual Sierpinski number is defined as an odd natural number k such that 2ⁿ + k is composite for all natural numbers n. There is a conjecture that the set of these numbers is the same as the set of Sierpinski numbers; for example, 2ⁿ + 78557 is composite for all natural numbers n.

The least n such that 2ⁿ + k is prime are (for odd ks)

1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 3, 2, 1, 1, 4, 2, 1, 2, 1, 1, 2, 1, 5, 2, 1, 3, 2, 1, 1, 8, 2, 1, 2, 1, 1, 4, 2, 1, 2, 1, 7, 2, 1, 3, 4, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 7, 4, 5, 3, 4, 2, 1, 2, 1, 3, 2, 1, 1, 10, 3, 3, 2, 1, 1, ... (sequence A067760 in the OEIS)

The odd ks which 2ⁿ + k are composite for all n < k are

773, 2131, 2491, 4471, 5101, 7013, 8543, 10711, 14717, 17659, 19081, 19249, 20273, 21661, 22193, 26213, 28433, ... (sequence A033919 in the OEIS)

There is also a "five or bust", similar to seventeen or bust, considers this problem, and found (probable) primes for all k < 78557 (the largest prime is 2^9092392 + 40291), so it is currently known that 78557 is the smallest dual Sierpinski number.

The following is a list of all those ks < 78557 and least n such that k2ⁿ + 1 is prime > 100000 or least n such that 2ⁿ + k is (probable) prime > 100000. (if both are > 100000, then the number k is colored red)

^*All numbers of the form 2ⁿ + k with n > 100000 are probable primes, i. e. not proved primes. All numbers of the form 2ⁿ + k with n < 100000 and 2^m + k is not prime of all m < n have been certified as definitely prime, the largest one is 2⁷³⁸⁴⁵ + 14717. At past, people want to prove the mixed Sierpinski problem, i. e. all odd numbers k < 78557 have a prime of the form k2ⁿ + 1 or 2ⁿ + k, to prove that 78557 is the smallest number with a covering set. In that time, all odd numbers k < 78557 are found a proved prime of at least one of these two forms except 19249, 28433, and 67607 (for 67607, only (in that time) strong probable prime 2¹⁶³⁸⁹ + 67607 is known), after that time it has been certified as definitely prime, so 67607 can be removed. Besides, a probable prime 2⁵⁵¹⁵⁴² + 19249 was found, since it is only a probable prime (and still a probable prime now!) and not a proved prime, we cannot actually say that 19249 can be removed. However, people want to find a prime or probable prime to the only remaining number, 28433, and a proved prime 28433×2^7830457 + 1 was found. Thus, 28433 can also be removed. After the large prime 19249×2^13018586 + 1 was found, the mixed Sierpinski problem is a theorem!

Sierpinski number base b

One can generalize the Sierpinski problem to an integer base b ≥ 2. A Sierpinski number base b is a positive integer k such that gcd(k + 1, b − 1) = 1. (if gcd(k + 1, b − 1) > 1, then gcd(k + 1, b − 1) is a trivial factor of k×bⁿ + 1 (Definition of trivial factors for the conjectures: Each and every n-value has the same factor)) For every integer b ≥ 2, there are infinitely many Sierpinski numbers base b.

Example 1: All numbers k congruent to 174308 mod 10124569 and not congruent to 4 mod 5 are Sierpinski numbers base 6, because of the covering set {7, 13, 31, 37, 97}. Besides, these k are not trivial since gcd(k + 1, 6 − 1) = 1 for these k. (The Sierpinski base 6 conjecture is not proven, it has 16 remaining k, see the list below)

Example 2: 4 is a Sierpinski number to all bases b congruent to 14 mod 15, because if b is congruent to 14 mod 15, then 4×bⁿ + 1 is divisible by 5 for all even n and divisible by 3 for all odd n. Besides, 4 is not a trivial k in these bases b since gcd(4 + 1, b − 1) = 1 for these bases b.

Example 3: 16 is a Sierpinski number in base 200, because if n is odd, then 16×200ⁿ + 1 is divisible by 3, and if n is congruent to 0 mod 4, then 16×200ⁿ + 1 is divisible by 17. Besides, if n is congruent to 2 mod 5, then 16×200ⁿ + 1 has algebraic factors. (The Sierpinski base 200 conjecture is not proven, it has one remaining k, namely 1)

Example 4: If k is between a multiple of 3 and a multiple of 13, then k×311ⁿ + 1 is divisible by either 3 or 13 for all positive integer n. The first few such k are 14, 25, 53, 64, 92, 103, 131, 142, ... However, all these k < 142 are also trivial k (i. e. gcd(k + 1, 311 − 1) is not 1). Thus, the smallest Sierpinski number base 311 is 142. (The Sierpinski base 311 conjecture is proven)

Example 5: If k is cube, then k×343ⁿ + 1 has algebraic factors for all positive integer n. The first few positive cubes are 1, 8, 27, 64, ... However, all these k < 64 are also trivial k (i. e. gcd(k + 1, 343 − 1) is not 1). Thus, the smallest Sierpinski number base 343 is 64. (The Sierpinski base 343 conjecture is proven)

We want to find and prove the smallest Sierpinski number base b for every integer b ≥ 2. It is a conjecture that if k is a Sierpinski number base b, then at least one of the three conditions holds:

1. All numbers of the form k×bⁿ + 1 have a factor in some covering set. (For example, b = 22, k = 6694, then all numbers of the form k×bⁿ + 1 have a factor in the covering set: {5, 23, 97})
2. k×bⁿ + 1 has algebraic factors. (For example, b = 16, k = 2500, then k×bⁿ + 1 can be factored to (50×4ⁿ − 10×2ⁿ + 1) × (50×4ⁿ + 10×2ⁿ + 1))
3. For some n, numbers of the form k×bⁿ + 1 have a factor in some covering set; and for all other n, k×bⁿ + 1 has algebraic factors. (For example, b = 55, k = 2500, then if n is not divisible by 4, then all numbers of the form k×bⁿ + 1 have a factor in the covering set: {7, 17}, if n is divisible by 4, then k×bⁿ + 1 can be factored to (50×55^n/2 − 10×55^n/4 + 1) × (50×55^n/2 + 10×55^n/4 + 1))

There is a special and interesting counterexample: b = 128, k = 8 (8 is not the smallest Sierpinski number base 128, the smallest Sierpinski number base 128 is 1 since 1×128ⁿ + 1 = (1×2ⁿ + 1) × (1×64ⁿ − 1×32ⁿ + 1×16ⁿ − 1×8ⁿ + 1×4ⁿ − 1×2ⁿ + 1), it has algebraic factors). Since 8×128ⁿ + 1 = 2⁷ⁿ⁺³ + 1, and if 2^r + 1 is prime, then r must be a power of 2, but 7n+3 cannot be a power of 2 since all powers of 2 are congruent to 1, 2, or 4 (mod 7), so all numbers of the form 8×128ⁿ + 1 are composite. Thus, 8 is a Sierpinski number base 128 since 8+1 and 128−1 are coprime. However, there is no covering set for 8×128ⁿ + 1 since if so, then we find the orders of 2 to mod all the primes in the covering set and find the exponents of highest power of 2 dividing the orders, and choose r greater than the largest exponent, since for every natural number r, there is a n such that 2^r divides 7n+3, so no prime in the covering set divide 2⁷ⁿ⁺³ (since if so, then the order of 2 to mod the prime is divisible by 2^r, but according above, the order of 2 to mod all primes in the covering set is not divisible by 2^r). Besides, 8×128ⁿ + 1 has no algebraic factors since there is no odd r > 1 such that both 128 and 8 are perfect rth powers, and 128 is not a perfect fourth power. Thus, this conjecture is not completely true, but it may be true except when b = a^r and k = a^s with even positive integer a not of the form m^t with integer m and odd integer t > 1, positive integer r and nonnegative integer s, gcd(r, s) = largest power of 2 dividing r, and 2^x ≡ s (mod r) has no solution.

In the following list, we only consider those positive integers k such that gcd(k + 1, b − 1) = 1, and all integer n must be ≥ 1.

Note: k-values that are a multiple of b and where k+1 is not prime are included in the conjectures (and included in the remaining k with red color if no primes are known for these k-values) but excluded from testing (Thus, never be the k of "largest 5 primes found"), since such k-values will have the same prime as k / b.

Conjectured smallest Sierpinski number base n are (start with n = 2)

78557, 125050976086, 66741, 159986, 174308, 1112646039348, 1, 2344, 9175, 1490, 521, 132, 4, 91218919470156, 2500, 278, 398, 765174, 8, 1002, 6694, 182, 30651, 262638, 221, 8, 4554, 4, 867, 6360528, 1, 1854, 6, 214018, 1886, 2604, 14, 166134, 826477, 8, 13372, 2256, 4, 53474, 14992, 8, 1219, 2944, 16, 5183582, 28674, 1966, 21, 2500, 20, 1188, 43071, 4, 16957, 15168, 8, 3511808, 1, ... (sequence A123159 in the OEIS)