Sierpinski number

In number theory, a Sierpinski or Sierpiński number is an odd natural number k such that $k\times 2^{n}+1$ 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:

\left\{\,k\cdot {}2^{n}+1:n\in \mathbb {N} \,\right\}.

Numbers in such a set with odd k and k < 2ⁿ are Proth numbers.

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 ${3, 5, 7, 13, 17, 241$ }. 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 $t 4 \cdot2 4n+2 + 1 = (t 2 \cdot2 2n+1 + t \cdot2 n+1 + 1)\cdot(t 2 \cdot2 2n+1 - t \cdot2 n+1 + 1)$ . This establishes that all $n \equiv 2 (mod 4)$ give rise to a composite, and so it remains to eliminate only $n \equiv 0, 1, 3 (mod 4)$ using a covering set.^[2]

Sierpiński problem

Unsolved problem in mathematics:

Is 78,557 the smallest Sierpiński number?

(more unsolved problems in mathematics)

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.^[1] As of August 2017, there are only five candidates:

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

which have not been eliminated as possible Sierpiński numbers.^[3]

Prime Sierpiński problem

In 1976, Nathan Mendelsohn determined that the second provable Sierpiński number is the prime k = 271129. The prime Sierpiński problem asks "what is the smallest prime Sierpiński number", and there is an ongoing "Prime Sierpiński search" which tries to prove that 271129 is the first Sierpiński number which is also a prime. The prime values of k less than 271129 for which a prime of the form k2ⁿ + 1 is not known (as of September 2017) are:

k = 22699, 67607, 79309, 79817, 152267, 156511, 222113, 225931, 237019.

The first two, being less than 78557, are also unsolved cases of the (non-prime) Sierpiński problem described above.

Extended Sierpiński problem

Suppose that both preceding Sierpiński problems had finally been solved, showing that 78557 is the smallest Sierpiński number and that 271129 is the smallest prime Sierpiński number. This still leaves unsolved the question of the second Sierpinski number; there could exist a composite Sierpiński number k such that $78557<k<271129$ . A 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.

Solving the extended Sierpiński problem, the most demanding of the three posed problems, requires the elimination of 23 remaining candidates $k<271129$ , of which nine are prime (see above) and fourteen are composite. The latter include k = 21181, 24737, 55459 from the original Sierpiński problem, unique to the extended Sierpiński problem. As of April 2018, the following ten values of k remain:

k = 91549, 99739, 131179, 163187, 200749, 202705, 209611, 227723, 229673, 238411.

The distributed volunteer computing project PrimeGrid is attempting to eliminate all the remaining values of k. As of April 2018, no prime has been found for these values of k with $n<11\,313\,676$ .^[4]

The most recent elimination was in April 2018, when $193997\times 2^{11452891}+1$ was found to be prime by PrimeGrid, eliminating k=193997. The number is 3,447,670 digits long.^[5]

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 (k ≤ 300), (301 ≤ k ≤ 600), (601 ≤ k ≤ 900), and (901 ≤ k ≤ 1200).

Simultaneously Sierpiński and Riesel

A number may be simultaneously Sierpiński and Riesel. These are called Brier numbers. The smallest five known examples are 3316923598096294713661, 10439679896374780276373, 11615103277955704975673, 12607110588854501953787, 17855036657007596110949, ... (A076335).^[6]

Dual Sierpinski problem

If we take the n of k2ⁿ + 1 to a negative integer, then the number become ${\frac {2^{-n}+k}{2^{-n}}}$ . 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^[7]), 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)

k	least n such that k2ⁿ + 1 is prime	least n such that 2ⁿ + k is (probable) prime
2131	44	4583176^*
4847	3321063	33
5359	5054502	170
7013	126113	104095^*
8543	5793	1191375^*
10223	31172165	19
17659	34	103766^*
19249	13018586	551542^*
21181	>29500000	28
22699	>29500000	26
24737	>29500000	17
25819	111842	70
27653	9167433	39
27923	158625	7
28433	7830457	2249255^*
33661	7031232	72
34999	462058	14
35461	4	139964^*
37967	23	308809^*
39781	176088	8
40291	8	9092392^*
41693	33	5146295^*
44131	995972	436
46157	698207	49
46187	104907	5
48527	951	105789^*
48833	167897	175
54767	1337287	5
55459	>29500000	14
59569	390454	26
60443	95901	148227^*
60451	44	983620^*
60541	176340	20
60947	783	176177^*
64133	161	304015^*
65567	1013803	5
67607	>29500000	16389
69109	1157446	26
74269	167546	22
75353	1	1518191^*

^*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 $k\times b^{n}+1$ (Definition of trivial factors for the conjectures: Each and every n-value has the same factor))^[8]^[9]^[10] 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 $16\times 200^{n}+1$ 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 4, then $16\times 200^{n}+1$ 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 $k\times 311^{n}+1$ 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 $k\times 343^{n}+1$ 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:

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

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 to the 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.

b	conjectured smallest Sierpinski k	covering set / algebraic factors	remaining k with no known primes (red ks indicate the k-values that are a multiple of b and where k+1 is not prime)	number of remaining k with no known primes (excluding the red ks)	testing limit of n (excluding the red ks)	largest 5 primes found (excluding the red ks)
2	78557	{3, 5, 7, 13, 19, 37, 73}	21181, 22699, 24737, 42362, 45398, 49474, 55459, 65536, 67607	6	k = 65536 at n = 2³³−17, other ks at n = 31M	10223×2^31172165+1 19249×2^13018586+1 27653×2^9167433+1 28433×2^7830457+1 33661×2^7031232+1
3	125050976086	{5, 7, 13, 17, 19, 37, 41, 193, 757}	6363484, 8911036, 12663902, 14138648, 14922034, 18302632, 19090452, 21497746, 23896396, 24019448, 24677704, 26733108, 33224138, 33381178, 35821276, 37063498, 37991706, 39431872, 42415944, 44766102, 46891088, 47628292, 54503602, 54907896, 56882284, 57271356, 60581468, 61270336, 63362504, 64493238, 69487006, 70143826, 70258712, 71689188, 72058344, 73440644, 74033112, 76020188, 77475694, 77574956, 78703468, 80199324, 81095362, 82085494, 88091054, 93455522, 96103394, 97696726, 99613186, 99672414, ...	69363 ks ≤ 13G	k ≤ 1G at n = 500K, 1G < k ≤ 13G at n = 25K	608558012×3⁴⁹⁸⁰⁹⁴+1 961852454×3⁴⁹⁵³⁷¹+1 98392246×3⁴⁹⁴⁵⁶⁴+1 138809722×3⁴⁹¹³¹⁸+1 965278456×3⁴⁸⁸¹⁵³+1
4	66741	{5, 7, 13, 17, 241}	18534, 21181, 22699, 49474, 55459, 64494, 65536	7	k = 65536 at n = 2³²−9, k = 18534 at n = 3.5M, k = 64494 at n = 2.4M, other ks at n = 15.5M	20446×4^15586082+1 19249×4^6509293+1 55306×4^4583716+1 56866×4^3915228+1 33661×4^3515616+1
5	159986	{3, 7, 13, 31, 601}	6436, 7528, 10918, 26798, 29914, 31712, 32180, 36412, 37640, 41738, 44348, 44738, 45748, 51208, 54590, 58642, 60394, 62698, 64258, 67612, 67748, 71492, 74632, 76724, 83936, 84284, 90056, 92906, 93484, 105464, 118568, 126134, 133990, 138514, 139196, 149570, 152588, 158560	32	PrimeGrid is currently searching at n > 2.5M	81556×5^2539960+1 92158×5^2145024+1 77072×5^2139921+1 154222×5^2091432+1 144052×5^2018290+1
6	174308	{7, 13, 31, 37, 97}	1296, 7776, 13215, 14505, 46656, 50252, 76441, 79290, 87030, 87800, 97131, 112783, 124125, 127688, 166753, 168610	12	k = 1296 at n = 2²⁸−5, other ks at n = 2M	139413×6^1279992+1 33706×6⁹¹⁰⁴⁶²+1 125098×6⁸⁹⁶⁶⁹⁶+1 31340×6⁸³³⁰⁹⁶+1 59506×6⁷⁸⁰⁸⁷⁷+1
7	1112646039348	{5, 13, 19, 43, 73, 181, 193, 1201}	987144, 1613796, 1911142, 2052426, 2471044, 3778846, 4023946, 4300896, 4369704, 4455408, 4723986, 4783794, 4810884, 6551056, 6910008, 7115518, 7248984, 8186656, 8566504, 9230674, 9284172, 9566736, ...	731 ks ≤ 50M	k ≤ 10M at n = 300K, 10M < k ≤ 50M at n = 25K	1952376×7²⁹³³⁵²+1 5452324×7²⁷⁷⁰⁹⁴+1 5071026×7²⁶¹⁹²¹+1 4325044×7²⁶⁰⁷¹³+1 4377694×7²⁴²³⁶⁵+1
8	1	1×8ⁿ + 1 = (1×2ⁿ + 1) × (1×4ⁿ − 1×2ⁿ + 1)	none (proven)	0	−	(none)
9	2344	{5, 7, 13, 73}	2036	1	2M	1846×9⁶⁵³⁷⁶+1 1804×9⁴⁴¹⁰³+1 1884×9¹⁶⁰⁹³+1 1306×9³³⁷⁴+1 914×9¹⁸¹³+1
10	9175	{7, 11, 13, 37}	100, 1000, 7666	2	k = 100 at n = 2²⁵−3, k = 7666 at n = 1.96M	5028×10⁸³⁹⁸²+1 7404×10⁴⁴⁸²⁶+1 8194×10²¹¹²⁹+1 4069×10¹²⁰⁹⁵+1 7809×10¹¹⁷⁹³+1
11	1490	{3, 7, 19, 37}	none (proven)	0	−	958×11³⁰⁰⁵⁴⁴+1 1468×11²⁶²⁵⁸+1 416×11¹²⁷⁴¹+1 1046×11³²⁰¹+1 1420×11²⁵⁶⁴+1
12	521	{5, 13, 29}	12, 144	1	2²⁵−2	404×12⁷¹⁴⁵⁵⁸+1 378×12²³⁸⁸+1 261×12⁶⁴⁴+1 407×12³⁶⁷+1 354×12²⁹¹+1
13	132	{5, 7, 17}	none (proven)	0	−	48×13⁶²⁶⁷+1 120×13¹⁵⁵²+1 106×13⁵⁶+1 64×13²⁶+1 112×13¹²+1
14	4	{3, 5}	none (proven)	0	−	1×14²+1 3×14¹+1 2×14¹+1
15	91218919470156	{13, 17, 113, 211, 241, 1489, 3877}	215432, 424074, 685812, 1936420, 2831648, 3100818, 3231480, 3789018, ...	49 ks ≤ 10M	k ≤ 5M at n = 200K, 5M < k ≤ 10M at n = 25K	3859132×15¹⁹⁵⁵⁶³+1 1868998×15¹⁸⁶⁸¹⁴+1 734268×15¹⁸⁰⁵⁶⁵+1 4713672×15⁸³⁹⁶²+1 3429436×15⁷⁸⁸⁶⁷+1
16	2500	2500×16ⁿ + 1 = (50×4ⁿ − 10×2ⁿ + 1) × (50×4ⁿ + 10×2ⁿ + 1)	none (proven)	0	−	2158×16¹⁰⁹⁰⁵+1 186×16⁵²²⁹+1 798×16²¹⁸¹+1 766×16¹⁵⁹⁸+1 1762×16¹⁵⁴⁹+1
17	278	{3, 5, 29}	244	1	2M	262×17¹⁸⁶⁷⁶⁸+1 160×17¹⁶⁶⁰⁴⁸+1 92×17⁵¹³¹¹+1 88×17⁴⁸⁶⁸+1 10×17¹³⁵⁶+1
18	398	{5, 13, 19}	18, 324	1	2²⁴−2	122×18²⁹²³¹⁸+1 381×18²⁴¹⁰⁸+1 291×18²⁴¹⁵+1 37×18⁴⁵⁷+1 362×18²⁵⁸+1
19	765174	{5, 7, 13, 127, 769}	1446, 2526, 2716, 3714, 4506, 4614, 6796, 10776, 14556, 15394, 15396, 15616, 16246, 17596, 19014, 19906, 20326, 20364, 21696, 24754, 25474, 27474, 29746, 29896, 29956, 30196, 36534, 38356, 39126, 39276, 42934, 43986, 44106, 45216, 45846, 46174, 47994, 50124, 51604, 53014, 55516, 57544, 59214, 60874, 61536, 63766, 64426, 64654, 64686, 64956, 66316, 67054, 68136, 69114, 70566, 72384, 72774, 73824, 76326, 77764, 79594, 80856, 80914, 81786, 83434, 84184, 84276, 84324, 85614, 85704, 86446, 86634, 87666, 87786, 87994, 90016, 90024, 92056, 95136, 96864, 98014, 99124, ...	571	160K	434674×19¹⁶⁰⁷⁵⁵+1 190584×19¹⁵⁹²⁹⁷+1 246124×19¹⁵⁸⁷⁵³+1 110946×19¹⁵⁷²⁸⁶+1 623184×19¹⁵³⁷⁶⁹+1
20	8	{3, 7}	none (proven)	0	−	6×20¹⁵+1 7×20²+1 4×20²+1 1×20²+1 5×20¹+1
21	1002	{11, 13, 17}	none (proven)	0	−	118×21¹⁹⁸⁴⁹+1 922×21²³⁰+1 736×21²¹⁵+1 976×21⁸⁴+1 978×21⁴³+1
22	6694	{5, 23, 97}	22, 484, 5128	2	k = 22 at n = 2²⁴−2, k = 5128 at n = 2M	1611×22⁷³⁸⁹⁸⁸+1 1908×22³⁵⁵³¹³+1 4233×22³⁰⁴⁰⁴⁶+1 5659×22⁹⁷⁷⁵⁸+1 6462×22⁴⁵⁵⁰⁷+1
23	182	{3, 5, 53}	none (proven)	0	−	68×23³⁶⁵²³⁹+1 8×23¹¹⁹²¹⁵+1 122×23¹⁴⁰⁴⁹+1 124×23³¹¹⁸+1 154×23²⁸⁹⁸+1
24	30651	{5, 7, 13, 73, 79}	656, 1099, 1851, 1864, 2164, 2351, 2586, 3404, 3526, 3609, 3706, 3846, 4606, 4894, 5129, 5316, 5324, 5386, 5889, 5974, 7276, 7746, 7844, 8054, 8091, 8161, 8369, 9279, 9304, 9701, 9721, 10026, 10156, 10531, 11346, 12799, 12969, 12991, 13716, 13984, 15744, 15921, 17334, 17819, 17876, 18006, 18204, 18911, 19031, 19094, 20219, 20731, 21459, 21526, 22289, 22356, 22479, 23844, 23874, 23981, 24784, 25964, 26279, 26376, 26804, 27344, 28099, 28249, 29009, 29091, 29349, 29464, 29566, 29601, 29641	73	221K	27611×24²¹⁹⁹⁴⁶+1 29116×24²¹⁶⁹⁸⁸+1 29619×24²⁰⁴²⁰³+1 10216×24¹⁸³⁹¹⁶+1 29549×24¹⁸²¹⁰⁵+1
25	262638	{7, 13, 31, 601}	222, 5550, 6436, 7528, 10918, 12864, 13548, 15588, 18576, 29914, 35970, 36412, 45330, 45748, 51208, 57240, 58434, 58642, 60394, 62698, 64258, 65610, 66678, 67612, 74632, 75666, 76896, 81186, 81556, 82962, 86334, 90240, 91038, 93378, 93484, 94212, ...	85	ks which k ≡ 1 mod 3 and k < 159986 at n = 1.1M, other ks at n = 300K	92158×25^1072512+1 154222×25^1045716+1 144052×25^1009145+1 120160×25⁸⁸⁴¹²⁴+1 186460×25⁷⁴³⁹⁹⁴+1
26	221	{3, 7, 19, 37}	65, 155	2	560K	32×26³¹⁸⁰⁷¹+1 217×26¹¹⁴⁵⁴+1 95×26¹⁶⁸³+1 178×26¹¹⁵⁴+1 138×26⁸²⁷+1
27	8	8×27ⁿ + 1 = (2×3ⁿ + 1) × (4×9ⁿ − 2×3ⁿ + 1)	none (proven)	0	−	2×27²+1 6×27¹+1 4×27¹+1
28	4554	{5, 29, 157}	871, 4552	2	k = 871 at n = 806K, k = 4552 at n = 1M	3394×28⁴²⁷²⁶²+1 4233×28³³¹¹³⁵+1 2377×28¹⁰⁴⁶²¹+1 1291×28²²⁸¹¹+1 2203×28¹³⁹¹¹+1
29	4	{3, 5}	none (proven)	0	−	2×29¹+1
30	867	{7, 13, 19, 31}	278, 588	2	500K	699×30¹¹⁸³⁷+1 242×30⁵⁰⁶⁴+1 659×30⁴⁹³⁶+1 311×30¹⁷⁶⁰+1 559×30¹⁶⁵⁴+1
31	6360528	{7, 13, 19, 37, 331}	76848, 124812, 135682, 148260, 155140, 185778, 208008, 217608, 231096, 245200, 257206, 260662, 262842, 263328, 272410, 303628, 311566, 313650, 348262, 369370, 385978, 413706, 419470, 450178, 457090, 464608, 475266, 479038, 504990, 512518, 513850, 579630, 596916, 613456, 626176, 635188, 679542, 699676, 731128, 748726, 749416, 756192, 758568, 785110, 786232, 805470, 810930, 811078, 822918, 828856, 843246, 852370, 871356, 879126, 888850, 889992, 899802, 908296, 916648, 961032, 965238, 966820, 975300, 994746, 999598, ...	503	100K	3419662×31⁹⁷⁸²⁶+1 1751346×31⁹⁷³⁷⁸+1 2983422×31⁹⁷⁰²¹+1 3298528×31⁹⁶⁹⁵⁷+1 4238758×31⁹⁶⁸⁵⁹+1
32	1	1×32ⁿ + 1 = (1×2ⁿ + 1) × (1×16ⁿ − 1×8ⁿ + 1×4ⁿ − 1×2ⁿ + 1)	none (proven)	0	−	(none)

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)

References

1 2 Sierpinski number at The Prime Glossary
↑ Anatoly S. Izotov (1995). "Note on Sierpinski Numbers" (PDF). Fibonacci Quarterly. 33 (3): 206.
↑ Seventeen or Bust at PrimeGrid.
↑ "Extended Sierpinski Problem statistics". www.primegrid.com. Retrieved 6 April 2018.
↑ Zimmerman, Van (5 April 2018). "ESP Mega Prime!". www.primegrid.com. Retrieved 6 April 2018.
↑ Problem 29.- Brier Numbers
↑ PRP tops, search for 2^n+40291
↑ Sierpinski conjectures in bases up to 1030
↑ Sierpinski conjectures in bases which are power of 2 up to 1024
↑ Sierpinski conjectures in bases up to 100

External links

The Sierpinski problem: definition and status
The Prime Glossary: Sierpinski number
Weisstein, Eric W. "Sierpinski's composite number theorem". MathWorld.
The dual Sierpinski problem
List of primes of the form: k*2^n+1, k<300
The Prime Sierpinski Problem, a related question
Five or bust, a related question
Grime, Dr. James. "78557 and Proth Primes" (video). YouTube. Brady Haran. Retrieved 13 November 2017.

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[PG-1] 1 2 Sierpinski number at The Prime Glossary

[FQ-2] Anatoly S. Izotov (1995). "Note on Sierpinski Numbers" (PDF). Fibonacci Quarterly. 33 (3): 206.

[3] Seventeen or Bust at PrimeGrid.

[4] "Extended Sierpinski Problem statistics". www.primegrid.com. Retrieved 6 April 2018.

[5] Zimmerman, Van (5 April 2018). "ESP Mega Prime!". www.primegrid.com. Retrieved 6 April 2018.

[6] Problem 29.- Brier Numbers

[7] PRP tops, search for 2^n+40291

[8] Sierpinski conjectures in bases up to 1030

[9] Sierpinski conjectures in bases which are power of 2 up to 1024

[10] Sierpinski conjectures in bases up to 100

Sierpinski number

Known Sierpiński numbers

Sierpiński problem

Prime Sierpiński problem

Extended Sierpiński problem

Smallest n for which k×2ⁿ+1 is prime

Simultaneously Sierpiński and Riesel

Dual Sierpinski problem

Sierpinski number base b

See also

References

Further reading

External links

Sierpinski number

Known Sierpiński numbers

Sierpiński problem

Prime Sierpiński problem

Extended Sierpiński problem

Smallest n for which k×2n+1 is prime

Simultaneously Sierpiński and Riesel

Dual Sierpinski problem

Sierpinski number base b

See also

References

Further reading

External links

Smallest n for which k×2ⁿ+1 is prime