Mersenne prime


In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form for some integer. They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th century. If is a composite number then so is. Therefore, an equivalent definition of the Mersenne primes is that they are the prime numbers of the form for some prime.
The exponents which give Mersenne primes are 2, 3, 5, 7, 13, 17, 19, 31, ... and the resulting Mersenne primes are 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, ....
Numbers of the form without the primality requirement may be called Mersenne numbers. Sometimes, however, Mersenne numbers are defined to have the additional requirement that be prime.
The smallest composite Mersenne number with prime exponent n is.
Mersenne primes were studied in antiquity because of their close connection to perfect numbers: the Euclid–Euler theorem asserts a one-to-one correspondence between even perfect numbers and Mersenne primes.
, 51 Mersenne primes are known. The largest known prime number,, is a Mersenne prime. Since 1997, all newly found Mersenne primes have been discovered by the Great Internet Mersenne Prime Search, a distributed computing project.

About Mersenne primes

Many fundamental questions about Mersenne primes remain unresolved. It is not even known whether the set of Mersenne primes is finite or infinite. The Lenstra–Pomerance–Wagstaff conjecture asserts that there are infinitely many Mersenne primes and predicts their order of growth. It is also not known whether infinitely many Mersenne numbers with prime exponents are composite, although this would follow from widely believed conjectures about prime numbers, for example, the infinitude of Sophie Germain primes congruent to 3. For these primes, will divide, for example,,,,,,,, and . Since for these primes, is congruent to 7 mod 8, so 2 is a quadratic residue mod, and the multiplicative order of 2 mod must divide =. Since is a prime, it must be or 1. However, it cannot be 1 since and 1 has no prime factors, so it must be. Hence, divides and cannot be prime.
The first four Mersenne primes are,, and and because the first Mersenne prime starts at, all Mersenne primes are congruent to 3. Other than and, all other Mersenne numbers are also congruent to 3. Consequently, in the prime factorization of a Mersenne number there must be at least one prime factor congruent to 3.
A basic theorem about Mersenne numbers states that if is prime, then the exponent must also be prime. This follows from the identity
This rules out primality for Mersenne numbers with composite exponent, such as.
Though the above examples might suggest that is prime for all primes, this is not the case, and the smallest counterexample is the Mersenne number
The evidence at hand suggests that a randomly selected Mersenne number is much more likely to be prime than an arbitrary randomly selected odd integer of similar size. Nonetheless, prime values of appear to grow increasingly sparse as increases. For example, eight of the first 11 primes give rise to a Mersenne prime , while is prime for only 43 of the first two million prime numbers.
The lack of any simple test to determine whether a given Mersenne number is prime makes the search for Mersenne primes a difficult task, since Mersenne numbers grow very rapidly. The Lucas–Lehmer primality test is an efficient primality test that greatly aids this task, making it much easier to test the primality of Mersenne numbers than that of most other numbers of the same size. The search for the largest known prime has somewhat of a cult following. Consequently, a lot of computer power has been expended searching for new Mersenne primes, much of which is now done using distributed computing.
Arithmetic modulo a Mersenne number is particularly efficient on a binary computer, making them popular choices when a prime modulus is desired, such as the Park–Miller random number generator. To find a primitive polynomial of Mersenne number order requires knowing the factorization of that number, so Mersenne primes allow one to find to primitive polynomials of very high order. Such primitive trinomials are used in pseudorandom number generators with very large periods such as the Mersenne twister, generalized shift register and Lagged Fibonacci generators.

Perfect numbers

Mersenne primes are closely connected to perfect numbers. In the 4th century BC, Euclid proved that if is prime, then ) is a perfect number. In the 18th century, Leonhard Euler proved that, conversely, all even perfect numbers have this form. This is known as the Euclid–Euler theorem. It is unknown whether there are any odd perfect numbers.

History

Mersenne primes take their name from the 17th-century French scholar Marin Mersenne, who compiled what was supposed to be a list of Mersenne primes with exponents up to 257. The exponents listed by Mersenne were as follows:
His list replicated the known primes of his time with exponents up to 19. His next entry, 31, was correct, but the list then became largely incorrect, as Mersenne mistakenly included and and omitted,, and . Mersenne gave little indication how he came up with his list.
Édouard Lucas proved in 1876 that is indeed prime, as Mersenne claimed. This was the largest known prime number for 75 years, and the largest ever found by hand. was determined to be prime in 1883 by Ivan Mikheevich Pervushin, though Mersenne claimed it was composite, and for this reason it is sometimes called Pervushin's number. This was the second-largest known prime number, and it remained so until 1911. Lucas had shown another error in Mersenne's list in 1876. Without finding a factor, Lucas demonstrated that is actually composite. No factor was found until a famous talk by Frank Nelson Cole in 1903. Without speaking a word, he went to a blackboard and raised 2 to the 67th power, then subtracted one. On the other side of the board, he multiplied and got the same number, then returned to his seat without speaking. He later said that the result had taken him "three years of Sundays" to find. A correct list of all Mersenne primes in this number range was completed and rigorously verified only about three centuries after Mersenne published his list.

Searching for Mersenne primes

Fast algorithms for finding Mersenne primes are available, and as of 2019 the seven largest known prime numbers are Mersenne primes.
The first four Mersenne primes,, and were known in antiquity. The fifth,, was discovered anonymously before 1461; the next two were found by Pietro Cataldi in 1588. After nearly two centuries, was verified to be prime by Leonhard Euler in 1772. The next was, found by Édouard Lucas in 1876, then by Ivan Mikheevich Pervushin in 1883. Two more were found early in the 20th century, by R. E. Powers in 1911 and 1914, respectively.
The best method presently known for testing the primality of Mersenne numbers is the Lucas–Lehmer primality test. Specifically, it can be shown that for prime, is prime if and only if divides, where and for.
During the era of manual calculation, all the exponents up to and including 257 were tested with the Lucas–Lehmer test and found to be composite. A notable contribution was made by retired Yale physics professor Horace Scudder Uhler, who did the calculations for exponents 157, 167, 193, 199, 227, and 229. Unfortunately for those investigators, the interval they were testing contains the largest known relative gap between Mersenne primes: the next Mersenne prime exponent, 521, would turn out to be more than four times larger than the previous record of 127.
The search for Mersenne primes was revolutionized by the introduction of the electronic digital computer. Alan Turing searched for them on the Manchester Mark 1 in 1949, but the first successful identification of a Mersenne prime,, by this means was achieved at 10:00 pm on January 30, 1952 using the U.S. National Bureau of Standards Western Automatic Computer at the Institute for Numerical Analysis at the University of California, Los Angeles, under the direction of Lehmer, with a computer search program written and run by Prof. R. M. Robinson. It was the first Mersenne prime to be identified in thirty-eight years; the next one,, was found by the computer a little less than two hours later. Three more —,, and — were found by the same program in the next several months. is the first Mersenne prime that is titanic, is the first gigantic, and was the first megaprime to be discovered, being a prime with at least 1,000,000 digits. All three were the first known prime of any kind of that size. The number of digits in the decimal representation of equals, where denotes the floor function.
In September 2008, mathematicians at UCLA participating in the Great Internet Mersenne Prime Search won part of a $100,000 prize from the Electronic Frontier Foundation for their discovery of a very nearly 13-million-digit Mersenne prime. The prize, finally confirmed in October 2009, is for the first known prime with at least 10 million digits. The prime was found on a Dell OptiPlex 745 on August 23, 2008. This was the eighth Mersenne prime discovered at UCLA.
On April 12, 2009, a GIMPS server log reported that a 47th Mersenne prime had possibly been found. The find was first noticed on June 4, 2009, and verified a week later. The prime is. Although it is chronologically the 47th Mersenne prime to be discovered, it is smaller than the largest known at the time, which was the 45th to be discovered.
On January 25, 2013, Curtis Cooper, a mathematician at the University of Central Missouri, discovered a 48th Mersenne prime, , as a result of a search executed by a GIMPS server network.
On January 19, 2016, Cooper published his discovery of a 49th Mersenne prime, , as a result of a search executed by a GIMPS server network. This was the fourth Mersenne prime discovered by Cooper and his team in the past ten years.
On September 2, 2016, the Great Internet Mersenne Prime Search finished verifying all tests below M37,156,667, thus officially confirming its position as the 45th Mersenne prime.
On January 3, 2018, it was announced that Jonathan Pace, a 51-year-old electrical engineer living in Germantown, Tennessee, had found a 50th Mersenne prime, , as a result of a search executed by a GIMPS server network.
On December 21, 2018, it was announced that The Great Internet Mersenne Prime Search discovered the largest known prime number,, having 24,862,048 digits. A computer volunteered by Patrick Laroche from Ocala, Florida made the find on December 7, 2018.

Theorems about Mersenne numbers

  1. If and are natural numbers such that is prime, then or.
  2. * Proof:. Then, so. Thus. However, is prime, so or. In the former case,, hence or In the latter case, or. If, however, which is not prime. Therefore,.
  3. If is prime, then is prime.
  4. * Proof: Suppose that is composite, hence can be written with and. Then so is composite. By contrapositive, if is prime then p is prime.
  5. If is an odd prime, then every prime that divides must be 1 plus a multiple of. This holds even when is prime.
  6. * For example, is prime, and. A composite example is, where and.
  7. * Proof: By Fermat's little theorem, is a factor of. Since is a factor of, for all positive integers, is also a factor of. Since is prime and is not a factor of, is also the smallest positive integer such that is a factor of. As a result, for all positive integers, is a factor of if and only if is a factor of. Therefore, since is a factor of, is a factor of so. Furthermore, since is a factor of, which is odd, is odd. Therefore,.
  8. * This fact leads to a proof of Euclid's theorem, which asserts the infinitude of primes, distinct from the proof written by Euclid: for every odd prime, all primes dividing are larger than ; thus there are always larger primes than any particular prime.
  9. * It follows from this fact that for every prime, there is at least one prime of the form less than or equal to, for some integer.
  10. If is an odd prime, then every prime that divides is congruent to.
  11. * Proof:, so is a square root of. By quadratic reciprocity, every prime modulus in which the number 2 has a square root is congruent to.
  12. A Mersenne prime cannot be a Wieferich prime.
  13. * Proof: We show if is a Mersenne prime, then the congruence does not hold. By Fermat's little theorem,. Therefore, one can write. If the given congruence is satisfied, then, therefore . Hence, and therefore. This leads to, which is impossible since.
  14. If and are natural numbers then and are coprime if and only if and are coprime. Consequently, a prime number divides at most one prime-exponent Mersenne number. That is, the set of pernicious Mersenne numbers is pairwise coprime.
  15. If and are both prime, and is congruent to, then divides.
  16. * Example: 11 and 23 are both prime, and, so 23 divides.
  17. * Proof: Let be. By Fermat's little theorem,, so either or. Supposing latter true, then, so −2 would be a quadratic residue mod. However, since is congruent to, is congruent to and therefore 2 is a quadratic residue mod. Also since is congruent to, −1 is a quadratic nonresidue mod, so −2 is the product of a residue and a nonresidue and hence it is a nonresidue, which is a contradiction. Hence, the former congruence must be true and divides.
  18. All composite divisors of prime-exponent Mersenne numbers are strong pseudoprimes to the base 2.
  19. With the exception of 1, a Mersenne number cannot be a perfect power. That is, and in accordance with Mihăilescu's theorem, the equation has no solutions where,, and are integers with and.

    List of known Mersenne primes

The table below lists all known Mersenne primes and in OEIS):
# digitsDiscoveredDiscovererMethod used
1231c. 430 BCAncient Greek mathematicians
2371c. 430 BCAncient Greek mathematicians
35312c. 300 BCAncient Greek mathematicians
471273c. 300 BCAncient Greek mathematicians
513819141456AnonymousTrial division
61713107161588Pietro CataldiTrial division
71952428761588Pietro CataldiTrial division
8312147483647101772Leonhard EulerTrial division with modular restrictions
9612305843009213693951191883 NovemberIvan M. PervushinLucas sequences
1089618970019642...137449562111271911 JuneRalph Ernest PowersLucas sequences
11107162259276829...578010288127331914 June 1Ralph Ernest PowersLucas sequences
12127170141183460...715884105727391876 January 10Édouard LucasLucas sequences
13521686479766013...2911150571511571952 January 30Raphael M. RobinsonLLT / SWAC
14607531137992816...2190317281271831952 January 30Raphael M. RobinsonLLT / SWAC
151,279104079321946...7031687290873861952 June 25Raphael M. RobinsonLLT / SWAC
162,203147597991521...6866977710076641952 October 7Raphael M. RobinsonLLT / SWAC
172,281446087557183...4181328363516871952 October 9Raphael M. RobinsonLLT / SWAC
183,217259117086013...3629093150719691957 September 8Hans RieselLLT / BESK
194,253190797007524...8153504849911,2811961 November 3Alexander HurwitzLLT / IBM 7090
204,423285542542228...9026085806071,3321961 November 3Alexander HurwitzLLT / IBM 7090
219,689478220278805...8262257541112,9171963 May 11Donald B. GilliesLLT / ILLIAC II
229,941346088282490...8837894635512,9931963 May 16Donald B. GilliesLLT / ILLIAC II
2311,213281411201369...0876963921913,3761963 June 2Donald B. GilliesLLT / ILLIAC II
2419,937431542479738...0309680414716,0021971 March 4Bryant TuckermanLLT / IBM 360/91
2521,701448679166119...3535118827516,5331978 October 30Landon Curt Noll & Laura NickelLLT / CDC Cyber 174
2623,209402874115778...5237792645116,9871979 February 9Landon Curt NollLLT / CDC Cyber 174
2744,497854509824303...96101122867113,3951979 April 8Harry L. Nelson & David SlowinskiLLT / Cray 1
2886,243536927995502...70943343820725,9621982 September 25David SlowinskiLLT / Cray 1
29110,503521928313341...08346551500733,2651988 January 29Walter Colquitt & Luke WelshLLT / NEC SX-2
30132,049512740276269...45573006131139,7511983 September 19David SlowinskiLLT / Cray X-MP
31216,091746093103064...10381552844765,0501985 September 1David SlowinskiLLT / Cray X-MP/24
32756,839174135906820...328544677887227,8321992 February 17David Slowinski & Paul GageLLT / Harwell Lab's Cray-2
33859,433129498125604...243500142591258,7161994 January 4David Slowinski & Paul GageLLT / Cray C90
341,257,787412245773621...976089366527378,6321996 September 3David Slowinski & Paul GageLLT / Cray T94
351,398,269814717564412...868451315711420,9211996 November 13GIMPS / Joel ArmengaudLLT / Prime95 on 90 MHz Pentium
362,976,221623340076248...743729201151895,9321997 August 24GIMPS / Gordon SpenceLLT / Prime95 on 100 MHz Pentium
373,021,377127411683030...973024694271909,5261998 January 27GIMPS / Roland ClarksonLLT / Prime95 on 200 MHz Pentium
386,972,593437075744127...1429241937912,098,9601999 June 1GIMPS / Nayan HajratwalaLLT / Prime95 on 350 MHz Pentium II IBM Aptiva
3913,466,917924947738006...4702562590714,053,9462001 November 14GIMPS / Michael CameronLLT / Prime95 on 800 MHz Athlon T-Bird
4020,996,011125976895450...7628556820476,320,4302003 November 17GIMPS / Michael ShaferLLT / Prime95 on 2 GHz Dell Dimension
4124,036,583299410429404...8827339694077,235,7332004 May 15GIMPS / Josh FindleyLLT / Prime95 on 2.4 GHz Pentium 4
4225,964,951122164630061...2805770772477,816,2302005 February 18GIMPS / Martin NowakLLT / Prime95 on 2.4 GHz Pentium 4
4330,402,457315416475618...4116529438719,152,0522005 December 15GIMPS / Curtis Cooper & Steven BooneLLT / Prime95 on 2 GHz Pentium 4
4432,582,657124575026015...1540539678719,808,3582006 September 4GIMPS / Curtis Cooper & Steven BooneLLT / Prime95 on 3 GHz Pentium 4
4537,156,667202254406890...02230822092711,185,2722008 September 6GIMPS / Hans-Michael ElvenichLLT / Prime95 on 2.83 GHz Core 2 Duo
4642,643,801169873516452...76556231475112,837,0642009 June 4GIMPS / Odd M. StrindmoLLT / Prime95 on 3 GHz Core 2
4743,112,609316470269330...16669715251112,978,1892008 August 23GIMPS / Edson SmithLLT / Prime95 on Dell Optiplex 745
4857,885,161581887266232...07172428595117,425,1702013 January 25GIMPS / Curtis CooperLLT / Prime95 on 3 GHz Intel Core2 Duo E8400
4974,207,281300376418084...39108643635122,338,6182016 January 7GIMPS / Curtis CooperLLT / Prime95 on Intel Core i7-4790
5077,232,917467333183359...06976217907123,249,4252017 December 26GIMPS / Jon PaceLLT / Prime95 on 3.3 GHz Intel Core i5-6600
5182,589,933148894445742...32521790259124,862,0482018 December 7GIMPS / Patrick LarocheLLT / Prime95 on Intel Core i5-4590T

All Mersenne numbers below the 51st Mersenne prime have been tested at least once but some have not been double-checked. Primes are not always discovered in increasing order. For example, the 29th Mersenne prime was discovered after the 30th and the 31st. Similarly, was followed by two smaller Mersenne primes, first 2 weeks later and then 9 months later. was the first discovered prime number with more than 10 million decimal digits.
The largest known Mersenne prime is also the largest known prime number.
The largest known prime has been a Mersenne prime since 1952, except between 1989 and 1992.

Factorization of composite Mersenne numbers

Since they are prime numbers, Mersenne primes are divisible only by 1 and by themselves. However, not all Mersenne numbers are Mersenne primes, and the composite Mersenne numbers may be factored non-trivially. Mersenne numbers are very good test cases for the special number field sieve algorithm, so often the largest number factorized with this algorithm has been a Mersenne number. , 2 − 1 is the record-holder, having been factored with a variant of the special number field sieve that allows the factorization of several numbers at once. See integer factorization records for links to more information. The special number field sieve can factorize numbers with more than one large factor. If a number has only one very large factor then other algorithms can factorize larger numbers by first finding small factors and then making a primality test on the cofactor., the largest factorization with probable prime factors allowed is, where is a 2,201,714-digit probable prime. It was discovered by Oliver Kruse. , the Mersenne number M1277 is the smallest composite Mersenne number with no known factors; it has no prime factors below 267.
The table below shows factorizations for the first 20 composite Mersenne numbers.
Factorization of
11204723 × 89
23838860747 × 178,481
29536870911233 × 1,103 × 2,089
37137438953471223 × 616,318,177
41219902325555113,367 × 164,511,353
438796093022207431 × 9,719 × 2,099,863
471407374883553272,351 × 4,513 × 13,264,529
5390071992547409916,361 × 69,431 × 20,394,401
5957646075230343487179,951 × 3,203,431,780,337
67147573952589676412927193,707,721 × 761,838,257,287
712361183241434822606847228,479 × 48,544,121 × 212,885,833
739444732965739290427391439 × 2,298,041 × 9,361,973,132,609
796044629098073145873530872,687 × 202,029,703 × 1,113,491,139,767
83967140655691...033397649407167 × 57,912,614,113,275,649,087,721
97158456325028...18708790067111,447 × 13,842,607,235,828,485,645,766,393
101253530120045...9934064107517,432,339,208,719 × 341,117,531,003,194,129
103101412048018...9736256430072,550,183,799 × 3,976,656,429,941,438,590,393
109649037107316...312041152511745,988,807 × 870,035,986,098,720,987,332,873
113103845937170...9926584401913,391 × 23,279 × 65,993 × 1,868,569 × 1,066,818,132,868,207
131272225893536...454145691647263 × 10,350,794,431,055,162,386,718,619,237,468,234,569

The number of factors for the first 500 Mersenne numbers can be found at.

Mersenne numbers in nature and elsewhere

In the mathematical problem Tower of Hanoi, solving a puzzle with an -disc tower requires steps, assuming no mistakes are made. The number of rice grains on the whole chessboard in the wheat and chessboard problem is.
The asteroid with minor planet number 8191 is named 8191 Mersenne after Marin Mersenne, because 8191 is a Mersenne prime.
In geometry, an integer right triangle that is primitive and has its even leg a power of 2 generates a unique right triangle such that its inradius is always a Mersenne number. For example, if the even leg is then because it is primitive it constrains the odd leg to be, the hypotenuse to be and its inradius to be.
The Mersenne numbers were studied with respect to the total number of accepting paths of non-deterministic polynomial time Turing machines in 2018 and intriguing inclusions were discovered.

Mersenne–Fermat primes

A Mersenne–Fermat number is defined as, with prime, natural number, and can be written as . When, it is a Mersenne number. When, it is a Fermat number. The only known Mersenne–Fermat primes with are
In fact,, where is the cyclotomic polynomial.

Generalizations

The simplest generalized Mersenne primes are prime numbers of the form, where is a low-degree polynomial with small integer coefficients. An example is, in this case,, and ; another example is, in this case,, and.
It is also natural to try to generalize primes of the form to primes of the form . However, is always divisible by, so unless the latter is a unit, the former is not a prime. This can be remedied by allowing b to be an algebraic integer instead of an integer:

Complex numbers

In the ring of integers, if is a unit, then is either 2 or 0. But are the usual Mersenne primes, and the formula does not lead to anything interesting. Thus, we can regard a ring of "integers" on complex numbers instead of real numbers, like Gaussian integers and Eisenstein integers.

Gaussian Mersenne primes

If we regard the ring of Gaussian integers, we get the case and, and can ask for which the number is a Gaussian prime which will then be called a Gaussian Mersenne prime.
is a Gaussian prime for the following :
Like the sequence of exponents for usual Mersenne primes, this sequence contains only prime numbers.
As for all Gaussian primes, the norms of these numbers are rational primes:

Eisenstein Mersenne primes

We can also regard the ring of Eisenstein integers, we get the case and, and can ask for what the number is an Eisenstein prime which will then be called a Eisenstein Mersenne prime.
is an Eisenstein prime for the following :
The norms of these Eisenstein primes are rational primes:

Divide an integer

Repunit primes

The other way to deal with the fact that is always divisible by, it is to simply take out this factor and ask which values of make
be prime. If, for example, we take, we get values of:
These primes are called repunit primes. Another example is when we take, we get values of:
It is a conjecture that for every integer which is not a perfect power, there are infinitely many values of such that is prime.
Least such that is prime are
For negative bases, they are
Least base such that is prime are
For negative bases, they are

Other generalized Mersenne primes

Another generalized Mersenne number is
with, any coprime integers, and. We can ask which makes this number prime. It can be shown that such must be primes themselves or equal to 4, and can be 4 if and only if and is prime. It is a conjecture that for any pair such that for every natural number, and are not both perfect th powers, and is not a perfect fourth power. there are infinitely many values of such that is prime. However, this has not been proved for any single value of.
numbers such that is prime
OEIS sequence
212, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 42643801, 43112609,..., 57885161,..., 74207281,..., 77232917,..., 82589933,...
2−13, 4*, 5, 7, 11, 13, 17, 19, 23, 31, 43, 61, 79, 101, 127, 167, 191, 199, 313, 347, 701, 1709, 2617, 3539, 5807, 10501, 10691, 11279, 12391, 14479, 42737, 83339, 95369, 117239, 127031, 138937, 141079, 267017, 269987, 374321, 986191, 4031399,..., 13347311, 13372531,...
322, 3, 5, 17, 29, 31, 53, 59, 101, 277, 647, 1061, 2381, 2833, 3613, 3853, 3929, 5297, 7417, 90217, 122219, 173191, 256199, 336353, 485977, 591827, 1059503,...
313, 7, 13, 71, 103, 541, 1091, 1367, 1627, 4177, 9011, 9551, 36913, 43063, 49681, 57917, 483611, 877843,...
3−12*, 3, 5, 7, 13, 23, 43, 281, 359, 487, 577, 1579, 1663, 1741, 3191, 9209, 11257, 12743, 13093, 17027, 26633, 104243, 134227, 152287, 700897, 1205459,...
3−23, 4*, 7, 11, 83, 149, 223, 599, 647, 1373, 8423, 149497, 388897,...
432, 3, 7, 17, 59, 283, 311, 383, 499, 521, 541, 599, 1193, 1993, 2671, 7547, 24019, 46301, 48121, 68597, 91283, 131497, 148663, 184463, 341233,...
412
4−12*, 3
4−33, 5, 19, 37, 173, 211, 227, 619, 977, 1237, 2437, 5741, 13463, 23929, 81223, 121271,...
543, 43, 59, 191, 223, 349, 563, 709, 743, 1663, 5471, 17707, 19609, 35449, 36697, 45259, 91493, 246497, 265007, 289937,...
5313, 19, 23, 31, 47, 127, 223, 281, 2083, 5281, 7411, 7433, 19051, 27239, 35863, 70327,...
522, 5, 7, 13, 19, 37, 59, 67, 79, 307, 331, 599, 1301, 12263, 12589, 18443, 20149, 27983,...
513, 7, 11, 13, 47, 127, 149, 181, 619, 929, 3407, 10949, 13241, 13873, 16519, 201359, 396413, 1888279,...
5−15, 67, 101, 103, 229, 347, 4013, 23297, 30133, 177337, 193939, 266863, 277183, 335429,...
5−22*, 3, 17, 19, 47, 101, 1709, 2539, 5591, 6037, 8011, 19373, 26489, 27427,...
5−32*, 3, 5, 7, 17, 19, 109, 509, 661, 709, 1231, 12889, 13043, 26723, 43963, 44789,...
5−44*, 5, 7, 19, 29, 61, 137, 883, 1381, 1823, 5227, 25561, 29537, 300893,...
652, 5, 11, 13, 23, 61, 83, 421, 1039, 1511, 31237, 60413, 113177, 135647, 258413,...
612, 3, 7, 29, 71, 127, 271, 509, 1049, 6389, 6883, 10613, 19889, 79987, 608099,...
6−12*, 3, 11, 31, 43, 47, 59, 107, 811, 2819, 4817, 9601, 33581, 38447, 41341, 131891, 196337,...
6−53, 4*, 5, 17, 397, 409, 643, 1783, 2617, 4583, 8783,...
762, 3, 7, 29, 41, 67, 1327, 1399, 2027, 69371, 86689, 355039,...
753, 5, 7, 113, 397, 577, 7573, 14561, 58543,...
742, 5, 11, 61, 619, 2879, 2957, 24371, 69247,...
733, 7, 19, 109, 131, 607, 863, 2917, 5923, 12421,...
723, 7, 19, 79, 431, 1373, 1801, 2897, 46997,...
715, 13, 131, 149, 1699, 14221, 35201, 126037, 371669, 1264699,...
7−13, 17, 23, 29, 47, 61, 1619, 18251, 106187, 201653,...
7−22*, 5, 23, 73, 101, 401, 419, 457, 811, 1163, 1511, 8011,...
7−33, 13, 31, 313, 3709, 7933, 14797, 30689, 38333,...
7−42*, 3, 5, 19, 41, 47, 8231, 33931, 43781, 50833, 53719, 67211,...
7−52*, 11, 31, 173, 271, 547, 1823, 2111, 5519, 7793, 22963, 41077, 49739,...
7−63, 53, 83, 487, 743,...
877, 11, 17, 29, 31, 79, 113, 131, 139, 4357, 44029, 76213, 83663, 173687, 336419, 615997,...
852, 19, 1021, 5077, 34031, 46099, 65707,...
832, 3, 7, 19, 31, 67, 89, 9227, 43891,...
813
8−12*
8−32*, 5, 163, 191, 229, 271, 733, 21059, 25237,...
8−52*, 7, 19, 167, 173, 223, 281, 21647,...
8−74*, 7, 13, 31, 43, 269, 353, 383, 619, 829, 877, 4957, 5711, 8317, 21739, 24029, 38299,...
982, 7, 29, 31, 67, 149, 401, 2531, 19913, 30773, 53857, 170099,...
973, 5, 7, 4703, 30113,...
953, 11, 17, 173, 839, 971, 40867, 45821,...
942
922, 3, 5, 13, 29, 37, 1021, 1399, 2137, 4493, 5521,...
91
9−13, 59, 223, 547, 773, 1009, 1823, 3803, 49223, 193247, 703393,...
9−22*, 3, 7, 127, 283, 883, 1523, 4001,...
9−42*, 3, 5, 7, 11, 17, 19, 41, 53, 109, 167, 2207, 3623, 5059, 5471, 7949, 21211, 32993, 60251,...
9−53, 5, 13, 17, 43, 127, 229, 277, 6043, 11131, 11821,...
9−72*, 3, 107, 197, 2843, 3571, 4451,..., 31517,...
9−83, 7, 13, 19, 307, 619, 2089, 7297, 75571, 76103, 98897,...
1092, 3, 7, 11, 19, 29, 401, 709, 2531, 15787, 66949, 282493,...
1072, 31, 103, 617, 10253, 10691,...
1032, 3, 5, 37, 599, 38393, 51431,...
1012, 19, 23, 317, 1031, 49081, 86453, 109297, 270343,...
10−15, 7, 19, 31, 53, 67, 293, 641, 2137, 3011, 268207,...
10−32*, 3, 19, 31, 101, 139, 167, 1097, 43151, 60703, 90499,...
10−72*, 3, 5, 11, 19, 1259, 1399, 2539, 2843, 5857, 10589,...
10−94*, 7, 67, 73, 1091, 1483, 10937,...
11103, 5, 19, 311, 317, 1129, 4253, 7699, 18199, 35153, 206081,...
1195, 31, 271, 929, 2789, 4153,...
1182, 7, 11, 17, 37, 521, 877, 2423,...
1175, 19, 67, 107, 593, 757, 1801, 2243, 2383, 6043, 10181, 11383, 15629,...
1162, 3, 11, 163, 191, 269, 1381, 1493,...
1155, 41, 149, 229, 263, 739, 3457, 20269, 98221,...
1143, 5, 11, 17, 71, 89, 827, 22307, 45893, 63521,...
1133, 5, 19, 31, 367, 389, 431, 2179, 10667, 13103, 90397,...
1122, 5, 11, 13, 331, 599, 18839, 23747, 24371, 29339, 32141, 67421,...
11117, 19, 73, 139, 907, 1907, 2029, 4801, 5153, 10867, 20161, 293831,...
11−15, 7, 179, 229, 439, 557, 6113, 223999, 327001,...
11−23, 5, 17, 67, 83, 101, 1373, 6101, 12119, 61781,...
11−33, 103, 271, 523, 23087, 69833,...
11−42*, 7, 53, 67, 71, 443, 26497,...
11−57, 11, 181, 421, 2297, 2797, 4129, 4139, 7151, 29033,...
11−62*, 5, 7, 107, 383, 17359, 21929, 26393,...
11−77, 1163, 4007, 10159,...
11−82*, 3, 13, 31, 59, 131, 223, 227, 1523,...
11−92*, 3, 17, 41, 43, 59, 83,...
11−1053, 421, 647, 1601, 35527,...
12112, 3, 7, 89, 101, 293, 4463, 70067,...
1272, 3, 7, 13, 47, 89, 139, 523, 1051,...
1252, 3, 31, 41, 53, 101, 421, 1259, 4721, 45259,...
1212, 3, 5, 19, 97, 109, 317, 353, 701, 9739, 14951, 37573, 46889, 769543,...
12−12*, 5, 11, 109, 193, 1483, 11353, 21419, 21911, 24071, 106859, 139739,...
12−52*, 3, 5, 13, 347, 977, 1091, 4861, 4967, 34679,...
12−72*, 3, 7, 67, 79, 167, 953, 1493, 3389, 4871,...
12−1147, 401, 509, 8609,...

*Note: if and is even, then the numbers are not included in the corresponding OEIS sequence.
A conjecture related to the generalized Mersenne primes:
For any integers and which satisfy the conditions:
  1. ,.
  2. and are coprime.
  3. For every natural number, and are not both perfect th powers.
  4. is not a perfect fourth power.
has prime numbers of the form
for prime, the prime numbers will be distributed near the best fit line
where
and there are about
prime numbers of this form less than.
We also have the following three properties:
  1. The number of prime numbers of the form less than or equal to is about.
  2. The expected number of prime numbers of the form with prime between and is about.
  3. The probability that number of the form is prime is about.
If this conjecture is true, then for all such pairs, let be the th prime of the form, the graph of versus is almost linear.
When, it is, a difference of two consecutive perfect th powers, and if is prime, then must be, because it is divisible by.
Least such that is prime are
Least such that is prime are

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