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Mersenne prime

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Template:Short description Template:Infobox integer sequence 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 Template:Math for some integer Template:Math. They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th century. If Template:Math is a composite number then so is Template:Math. Therefore, an equivalent definition of the Mersenne primes is that they are the prime numbers of the form Template:Math for some prime Template:Math.

The exponents Template:Math which give Mersenne primes are 2, 3, 5, 7, 13, 17, 19, 31, ... (sequence A000043 in the OEIS) and the resulting Mersenne primes are 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, ... (sequence A000668 in the OEIS).

Numbers of the form Template:Math without the primality requirement may be called Mersenne numbers. Sometimes, however, Mersenne numbers are defined to have the additional requirement that Template:Math should be prime. The smallest composite Mersenne number with prime exponent n is Template:Nowrap.

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. Many of the largest known primes are Mersenne primes because Mersenne numbers are easier to check for primality.

Template:As of, 52 Mersenne primes are known. The largest known prime number, Template:Nowrap, is a Mersenne prime.<ref name="GIMPS-2024">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref name="GIMPS-2018">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> Since 1997, all newly found Mersenne primes have been discovered by the Great Internet Mersenne Prime Search, a distributed computing project. In December 2020, a major milestone in the project was passed after all exponents below 100 million were checked at least once.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

About Mersenne primesEdit

Template:Unsolved 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 claims that there are infinitely many Mersenne primes and predicts their order of growth and frequency: For every number Template:Mvar, there should on average be about <math>e^\gamma\cdot\log_2(10) \approx 5.92</math> primes Template:Mvar with Template:Mvar decimal digits for which <math>M_p</math> is prime. Here, Template:Mvar is the Euler–Mascheroni constant.

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 (mod 4). For these primes Template:Mvar, Template:Math (which is also prime) will divide Template:Mvar, for example, Template:Math, Template:Math, Template:Math, Template:Math, Template:Math, Template:Math, Template:Math, and Template:Math (sequence A002515 in the OEIS). For these primes Template:Mvar, Template:Math is congruent to 7 mod 8, so 2 is a quadratic residue mod Template:Math, and the multiplicative order of 2 mod Template:Math must divide <math display="inline">\frac{(2p+1)-1}{2} = p</math>. Since Template:Mvar is a prime, it must be Template:Mvar or 1. However, it cannot be 1 since <math>\Phi_1(2) = 1</math> and 1 has no prime factors, so it must be Template:Mvar. Hence, Template:Math divides <math>\Phi_p(2) = 2^p-1</math> and <math>2^p-1 = M_p</math> cannot be prime. The first four Mersenne primes are Template:Math, Template:Math, Template:Math and Template:Math and because the first Mersenne prime starts at Template:Math, all Mersenne primes are congruent to 3 (mod 4). Other than Template:Math and Template:Math, all other Mersenne numbers are also congruent to 3 (mod 4). Consequently, in the prime factorization of a Mersenne number ( Template:Math ) there must be at least one prime factor congruent to 3 (mod 4).

A basic theorem about Mersenne numbers states that if Template:Mvar is prime, then the exponent Template:Math must also be prime. This follows from the identity <math display="block">\begin{align} 2^{ab}-1 &=(2^a-1)\cdot \left(1+2^a+2^{2a}+2^{3a}+\cdots+2^{(b-1)a}\right)\\ &=(2^b-1)\cdot \left(1+2^b+2^{2b}+2^{3b}+\cdots+2^{(a-1)b}\right). \end{align}</math> This rules out primality for Mersenne numbers with a composite exponent, such as Template:Math.

Though the above examples might suggest that Template:Mvar is prime for all primes Template:Mvar, this is not the case, and the smallest counterexample is the Mersenne number

Template:Math.

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.<ref name="Wagstaff">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> Nonetheless, prime values of Template:Mvar appear to grow increasingly sparse as Template:Mvar increases. For example, eight of the first 11 primes Template:Mvar give rise to a Mersenne prime Template:Mvar (the correct terms on Mersenne's original list), while Template:Mvar is prime for only 43 of the first two million prime numbers (up to 32,452,843).

Since Mersenne numbers grow very rapidly, the search for Mersenne primes is a difficult task, even though there is a simple efficient test to determine whether a given Mersenne number is prime: the Lucas–Lehmer primality test (LLT), which makes 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.Template:Citation needed Consequently, a large amount 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 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 numbersEdit

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Mersenne primes Template:Math are closely connected to perfect numbers. In the 4th century BC, Euclid proved that if Template:Math is prime, then Template:Math) is a perfect number. In the 18th century, Leonhard Euler proved that, conversely, all even perfect numbers have this form.<ref>Chris K. Caldwell, Mersenne Primes: History, Theorems and Lists</ref> This is known as the Euclid–Euler theorem. It is unknown whether there are any odd perfect numbers.

HistoryEdit

2 3 5 7 11 13 17 19
23 29 31 37 41 43 47 53
59 61 67 71 73 79 83 89
97 101 103 107 109 113 127 131
137 139 149 151 157 163 167 173
179 181 191 193 197 199 211 223
227 229 233 239 241 251 257 263
269 271 277 281 283 293 307 311
The first 64 prime exponents with those corresponding to Mersenne primes shaded in cyan and in bold, and those thought to do so by Mersenne in red and bold

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 in 1644 were as follows:

2, 3, 5, 7, 13, 17, 19, 31, 67, 127, 257.

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 Template:Math and Template:Math (which are composite) and omitted Template:Math, Template:Math, and Template:Math (which are prime). Mersenne gave little indication of how he came up with his list.<ref>The Prime Pages, Mersenne's conjecture.</ref>

Édouard Lucas proved in 1876 that Template:Math is indeed prime, as Mersenne claimed. This was the largest known prime number for 75 years until 1951, when Aimé Ferrier found a larger prime, <math>(2^{148}+1)/17</math>, using a desk calculating machine.<ref>Template:Cite book</ref>Template:Rp Template:Math 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 numberTemplate:Cn. 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 by demonstrating that Template:Math was composite without finding a factor. No factor was found until a famous talk by Frank Nelson Cole in 1903.<ref>Template:Cite journal</ref> Without speaking a word, he went to a blackboard and raised 2 to the 67th power, then subtracted one, resulting in the number Template:Nowrap. On the other side of the board, he multiplied Template:Nowrap and got the same number, then returned to his seat (to applause) without speaking.<ref>Template:Cite book p. 228.</ref> He later said that the result had taken him "three years of Sundays" to find.<ref>Template:Cite news</ref> 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 primesEdit

Fast algorithms for finding Mersenne primes are available, and Template:As of, the seven largest known prime numbers are Mersenne primes.

The first four Mersenne primes Template:Math, Template:Math, Template:Math and Template:Math were known in antiquity. The fifth, Template:Math, was discovered anonymously before 1461; the next two (Template:Math and Template:Math) were found by Pietro Cataldi in 1588. After nearly two centuries, Template:Math was verified to be prime by Leonhard Euler in 1772. The next (in historical, not numerical order) was Template:Math, found by Édouard Lucas in 1876, then Template:Math by Ivan Mikheevich Pervushin in 1883. Two more (Template:Math and Template:Math) were found early in the 20th century, by R. E. Powers in 1911 and 1914, respectively.

The most efficient method presently known for testing the primality of Mersenne numbers is the Lucas–Lehmer primality test. Specifically, it can be shown that for prime Template:Math, Template:Math is prime if and only if Template:Math divides Template:Math, where Template:Math and Template:Math for Template:Math.

During the era of manual calculation, all previously untested 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.<ref>Template:Cite journal</ref> 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 as large as the previous record of 127.

File:Digits in largest prime found as a function of time.svg
Graph of number of digits in largest known Mersenne prime by year – electronic era. The vertical scale is logarithmic in the number of digits, thus being a <math>\log(\log(y))</math> function in the value of the prime.

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,<ref>Brian Napper, The Mathematics Department and the Mark 1.</ref> but the first successful identification of a Mersenne prime, Template:Math, by this means was achieved at 10:00 pm on January 30, 1952, using the U.S. National Bureau of Standards Western Automatic Computer (SWAC) at the Institute for Numerical Analysis at the University of California, Los Angeles (UCLA), under the direction of D. H. 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, Template:Math, was found by the computer a little less than two hours later. Three more — Template:Math, Template:Math, and Template:Math — were found by the same program in the next several months. Template:Math was the first prime discovered with more than 1000 digits, Template:Math was the first with more than 10,000, and Template:Math was the first with more than a million. In general, the number of digits in the decimal representation of Template:Math equals Template:Math, where Template:Math denotes the floor function (or equivalently Template:Math).

In September 2008, mathematicians at UCLA participating in the Great Internet Mersenne Prime Search (GIMPS) 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.<ref>Template:Cite news</ref>

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 Template:Nowrap. 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, Template:Nowrap (a number with 17,425,170 digits), as a result of a search executed by a GIMPS server network.<ref>Template:Cite magazine</ref>

On January 19, 2016, Cooper published his discovery of a 49th Mersenne prime, Template:Nowrap (a number with 22,338,618 digits), as a result of a search executed by a GIMPS server network.<ref name="GIMPS-2016">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref>Template:Cite news</ref><ref name="NYT-20160121">Template:Cite news</ref> 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 Template:Math, thus officially confirming its position as the 45th Mersenne prime.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

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, Template:Nowrap (a number with 23,249,425 digits), as a result of a search executed by a GIMPS server network.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> The discovery was made by a computer in the offices of a church in the same town.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

On December 21, 2018, it was announced that The Great Internet Mersenne Prime Search (GIMPS) discovered a new prime number, Template:Nowrap, having 24,862,048 digits. A computer volunteered by Patrick Laroche from Ocala, Florida made the find on December 7, 2018.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

In late 2020, GIMPS began using a new technique to rule out potential Mersenne primes called the Probable prime (PRP) test, based on development from Robert Gerbicz in 2017, and a simple way to verify tests developed by Krzysztof Pietrzak in 2018. Due to the low error rate and ease of proof, this nearly halved the computing time to rule out potential primes over the Lucas-Lehmer test (as two users would no longer have to perform the same test to confirm the other's result), although exponents passing the PRP test still require one to confirm their primality.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

On October 12, 2024, a user named Luke Durant from San Jose, California, found the current largest known Mersenne prime, Template:Nowrap, having 41,024,320 digits. This marks the first Mersenne prime with an exponent surpassing 8 digits. This was announced on October 21, 2024.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

Theorems about Mersenne numbersEdit

Mersenne numbers are 0, 1, 3, 7, 15, 31, 63, ... (sequence A000225 in the OEIS).

  1. If Template:Math and Template:Mvar are natural numbers such that Template:Math is prime, then Template:Math or Template:Math.
  2. If Template:Math is prime, then Template:Mvar is prime.
  3. If Template:Mvar is an odd prime, then every prime Template:Mvar that divides Template:Math must be 1 plus a multiple of Template:Math. This holds even when Template:Math is prime.
  4. If Template:Mvar is an odd prime, then every prime Template:Mvar that divides Template:Math is congruent to Template:Nowrap.
  5. A Mersenne prime cannot be a Wieferich prime.
  6. If Template:Mvar and Template:Mvar are natural numbers then Template:Mvar and Template:Mvar are coprime if and only if Template:Math and Template:Math are coprime. Consequently, a prime number divides at most one prime-exponent Mersenne number.<ref>Will Edgington's Mersenne Page Template:Webarchive</ref> That is, the set of pernicious Mersenne numbers is pairwise coprime.
  7. If Template:Mvar and Template:Math are both prime (meaning that Template:Mvar is a Sophie Germain prime), and Template:Mvar is congruent to Template:Nowrap, then Template:Math divides Template:Math.<ref>{{#invoke:citation/CS1|citation

|CitationClass=web }}</ref>

  2. All composite divisors of prime-exponent Mersenne numbers are strong pseudoprimes to the base 2.
  3. With the exception of 1, a Mersenne number cannot be a perfect power. That is, and in accordance with Mihăilescu's theorem, the equation Template:Math has no solutions where Template:Mvar, Template:Mvar, and Template:Mvar are integers with Template:Math and Template:Math.
  4. The Mersenne number sequence is a member of the family of Lucas sequences. It is Template:Math(3, 2). That is, Mersenne number Template:Math with Template:Math and Template:Math.

List of known Mersenne primesEdit

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Template:As of, the 52 known Mersenne primes are 2p − 1 for the following p:

2, 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, 136279841. (sequence A000043 in the OEIS)

Factorization of composite Mersenne numbersEdit

Since they are prime numbers, Mersenne primes are divisible only by 1 and themselves. However, not all Mersenne numbers are Mersenne primes. 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. Template:As of, Template:Math is the record-holder,<ref>Template:Cite book</ref> 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 running a primality test on the cofactor. Template:As of, the largest completely factored number (with probable prime factors allowed) is Template:Math, where Template:Mvar is a 3,829,294-digit probable prime. It was discovered by a GIMPS participant with nickname "Funky Waddle".<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> Template:As of, the Mersenne number M1277 is the smallest composite Mersenne number with no known factors; it has no prime factors below 268,<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> and is very unlikely to have any factors below 1065 (~2216).<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

The table below shows factorizations for the first 20 composite Mersenne numbers where the exponent Template:Math is a prime number (sequence A244453 in the OEIS).

Template:Math Template:Math Digits Factorization of Template:Math
11 2047 4 23 × 89
23 8388607 7 47 × 178,481
29 536870911 9 233 × 1,103 × 2,089
37 137438953471 12 223 × 616,318,177
41 2199023255551 13 13,367 × 164,511,353
43 8796093022207 13 431 × 9,719 × 2,099,863
47 140737488355327 15 2,351 × 4,513 × 13,264,529
53 9007199254740991 16 6,361 × 69,431 × 20,394,401
59 576460752303423487 18 179,951 × 3,203,431,780,337 (13 digits)
67 147573952589676412927 21 193,707,721 × 761,838,257,287 (12 digits)
71 2361183241434822606847 22 228,479 × 48,544,121 × 212,885,833
73 9444732965739290427391 22 439 × 2,298,041 × 9,361,973,132,609 (13 digits)
79 604462909807314587353087 24 2,687 × 202,029,703 × 1,113,491,139,767 (13 digits)
83 9671406556917033397649407 25 167 × 57,912,614,113,275,649,087,721 (23 digits)
97 158456325028528675187087900671 30 11,447 × 13,842,607,235,828,485,645,766,393 (26 digits)
101 2535301200456458802993406410751 31 7,432,339,208,719 (13 digits) × 341,117,531,003,194,129 (18 digits)
103 10141204801825835211973625643007 32 2,550,183,799 × 3,976,656,429,941,438,590,393 (22 digits)
109 649037107316853453566312041152511 33 745,988,807 × 870,035,986,098,720,987,332,873 (24 digits)
113 10384593717069655257060992658440191 35 3,391 × 23,279 × 65,993 × 1,868,569 × 1,066,818,132,868,207 (16 digits)
131 2722258935367507707706996859454145691647 40 263 × 10,350,794,431,055,162,386,718,619,237,468,234,569 (38 digits)

The number of factors for the first 500 Mersenne numbers can be found at (sequence A046800 in the OEIS).

Mersenne numbers in nature and elsewhereEdit

In the mathematical problem Tower of Hanoi, solving a puzzle with an Template:Math-disc tower requires Template:Math steps, assuming no mistakes are made.<ref>Template:Cite book</ref> The number of rice grains on the whole chessboard in the wheat and chessboard problem is Template:Math.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

The asteroid with minor planet number 8191 is named 8191 Mersenne after Marin Mersenne, because 8191 is a Mersenne prime.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

In geometry, an integer right triangle that is primitive and has its even leg a power of 2 ( Template:Math ) generates a unique right triangle such that its inradius is always a Mersenne number. For example, if the even leg is Template:Math then because it is primitive it constrains the odd leg to be Template:Math, the hypotenuse to be Template:Math and its inradius to be Template:Math.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

Mersenne–Fermat primesEdit

A Mersenne–Fermat number is defined as Template:Math with Template:Math prime, Template:Math natural number, and can be written as Template:Math. When Template:Math, it is a Mersenne number. When Template:Math, it is a Fermat number. The only known Mersenne–Fermat primes with Template:Math are

Template:Math and Template:Math.<ref>{{#invoke:citation/CS1|citation

|CitationClass=web }}</ref>

In fact, Template:Math, where Template:Math is the cyclotomic polynomial.

GeneralizationsEdit

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} The simplest generalized Mersenne primes are prime numbers of the form Template:Math, where Template:Math is a low-degree polynomial with small integer coefficients.<ref>Template:Cite book</ref> An example is Template:Math, in this case, Template:Math, and Template:Math; another example is Template:Math, in this case, Template:Math, and Template:Math.

It is also natural to try to generalize primes of the form Template:Math to primes of the form Template:Math (for Template:Math and Template:Math). However (see also theorems above), Template:Math is always divisible by Template:Math, 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 numbersEdit

In the ring of integers (on real numbers), if Template:Math is a unit, then Template:Math is either 2 or 0. But Template:Math are the usual Mersenne primes, and the formula Template:Math does not lead to anything interesting (since it is always −1 for all Template:Math). Thus, we can regard a ring of "integers" on complex numbers instead of real numbers, like Gaussian integers and Eisenstein integers.

Gaussian Mersenne primesEdit

If we regard the ring of Gaussian integers, we get the case Template:Math and Template:Math, and can ask (WLOG) for which Template:Math the number Template:Math is a Gaussian prime which will then be called a Gaussian Mersenne prime.<ref>Chris Caldwell: The Prime Glossary: Gaussian Mersenne (part of the Prime Pages)</ref>

Template:Math is a Gaussian prime for the following Template:Math:

2, 3, 5, 7, 11, 19, 29, 47, 73, 79, 113, 151, 157, 163, 167, 239, 241, 283, 353, 367, 379, 457, 997, 1367, 3041, 10141, 14699, 27529, 49207, 77291, 85237, 106693, 160423, 203789, 364289, 991961, 1203793, 1667321, 3704053, 4792057, ... (sequence A057429 in the OEIS)

Like the sequence of exponents for usual Mersenne primes, this sequence contains only (rational) prime numbers.

As for all Gaussian primes, the norms (that is, squares of absolute values) of these numbers are rational primes:

5, 13, 41, 113, 2113, 525313, 536903681, 140737471578113, ... (sequence A182300 in the OEIS).

Eisenstein Mersenne primesEdit

One may encounter cases where such a Mersenne prime is also an Eisenstein prime, being of the form Template:Math and Template:Math. In these cases, such numbers are called Eisenstein Mersenne primes.

Template:Math is an Eisenstein prime for the following Template:Math:

2, 5, 7, 11, 17, 19, 79, 163, 193, 239, 317, 353, 659, 709, 1049, 1103, 1759, 2029, 5153, 7541, 9049, 10453, 23743, 255361, 534827, 2237561, ... (sequence A066408 in the OEIS)

The norms (that is, squares of absolute values) of these Eisenstein primes are rational primes:

7, 271, 2269, 176419, 129159847, 1162320517, ... (sequence A066413 in the OEIS)

Divide an integerEdit

Repunit primesEdit

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} The other way to deal with the fact that Template:Math is always divisible by Template:Math, it is to simply take out this factor and ask which values of Template:Math make

<math>\frac{b^n-1}{b-1}</math>

be prime. (The integer Template:Math can be either positive or negative.) If, for example, we take Template:Math, we get Template:Math values of:

2, 19, 23, 317, 1031, 49081, 86453, 109297, 270343, ... (sequence A004023 in the OEIS),
corresponding to primes 11, 1111111111111111111, 11111111111111111111111, ... (sequence A004022 in the OEIS).

These primes are called repunit primes. Another example is when we take Template:Math, we get Template:Math values of:

2, 5, 11, 109, 193, 1483, 11353, 21419, 21911, 24071, 106859, 139739, ... (sequence A057178 in the OEIS),
corresponding to primes −11, 19141, 57154490053, ....

It is a conjecture that for every integer Template:Math which is not a perfect power, there are infinitely many values of Template:Math such that Template:Math is prime. (When Template:Math is a perfect power, it can be shown that there is at most one Template:Math value such that Template:Math is prime)

Least Template:Math such that Template:Math is prime are (starting with Template:Math, Template:Math if no such Template:Math exists)

2, 3, 2, 3, 2, 5, 3, 0, 2, 17, 2, 5, 3, 3, 2, 3, 2, 19, 3, 3, 2, 5, 3, 0, 7, 3, 2, 5, 2, 7, 0, 3, 13, 313, 2, 13, 3, 349, 2, 3, 2, 5, 5, 19, 2, 127, 19, 0, 3, 4229, 2, 11, 3, 17, 7, 3, 2, 3, 2, 7, 3, 5, 0, 19, 2, 19, 5, 3, 2, 3, 2, ... (sequence A084740 in the OEIS)

For negative bases Template:Math, they are (starting with Template:Math, Template:Math if no such Template:Math exists)

3, 2, 2, 5, 2, 3, 2, 3, 5, 5, 2, 3, 2, 3, 3, 7, 2, 17, 2, 3, 3, 11, 2, 3, 11, 0, 3, 7, 2, 109, 2, 5, 3, 11, 31, 5, 2, 3, 53, 17, 2, 5, 2, 103, 7, 5, 2, 7, 1153, 3, 7, 21943, 2, 3, 37, 53, 3, 17, 2, 7, 2, 3, 0, 19, 7, 3, 2, 11, 3, 5, 2, ... (sequence A084742 in the OEIS) (notice this OEIS sequence does not allow Template:Math)

Least base Template:Math such that Template:Math is prime are

2, 2, 2, 2, 5, 2, 2, 2, 10, 6, 2, 61, 14, 15, 5, 24, 19, 2, 46, 3, 11, 22, 41, 2, 12, 22, 3, 2, 12, 86, 2, 7, 13, 11, 5, 29, 56, 30, 44, 60, 304, 5, 74, 118, 33, 156, 46, 183, 72, 606, 602, 223, 115, 37, 52, 104, 41, 6, 338, 217, ... (sequence A066180 in the OEIS)

For negative bases Template:Math, they are

3, 2, 2, 2, 2, 2, 2, 2, 2, 7, 2, 16, 61, 2, 6, 10, 6, 2, 5, 46, 18, 2, 49, 16, 70, 2, 5, 6, 12, 92, 2, 48, 89, 30, 16, 147, 19, 19, 2, 16, 11, 289, 2, 12, 52, 2, 66, 9, 22, 5, 489, 69, 137, 16, 36, 96, 76, 117, 26, 3, ... (sequence A103795 in the OEIS)

Other generalized Mersenne primesEdit

Another generalized Mersenne number is

<math>\frac{a^n-b^n}{a-b}</math>

with Template:Mvar, Template:Mvar any coprime integers, Template:Math and Template:Math. (Since Template:Math is always divisible by Template:Math, the division is necessary for there to be any chance of finding prime numbers.)Template:Efn We can ask which Template:Mvar makes this number prime. It can be shown that such Template:Mvar must be primes themselves or equal to 4, and Template:Mvar can be 4 if and only if Template:Math and Template:Math is prime.Template:Efn It is a conjecture that for any pair Template:Math such that Template:Mvar and Template:Mvar are not both perfect Template:Mvarth powers for any Template:Mvar and Template:Math is not a perfect fourth power, there are infinitely many values of Template:Mvar such that Template:Math is prime.Template:Efn However, this has not been proved for any single value of Template:Math.

For more information, see <ref>Template:Math and Template:Math for Template:Mvar = 2 to 50</ref><ref>Template:Math for Template:Mvar = 2 to 160</ref><ref>Template:Math for Template:Mvar = 2 to 160</ref><ref>Template:Math for Template:Mvar = 1 to 160</ref><ref>Template:Math for Template:Mvar = 1 to 40</ref><ref>Template:Math for odd Template:Mvar = 1 to 107</ref><ref>Template:Math for Template:Mvar = 2 to 200</ref><ref>PRP records, search for Template:Tmath, that is, Template:Math</ref><ref>PRP records, search for Template:Tmath, that is, Template:Math</ref>
Template:Nobold Template:Nobold numbers Template:Nobold such that Template:Nobold is prime
(some large terms are only probable primes, these Template:Nobold are checked up to Template:Nobold for Template:Nobold or Template:Nobold, Template:Nobold for Template:Nobold) 		OEIS sequence
2 1 2, 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, ..., 136279841, ... A000043
2 −1 3, 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, ... A000978
3 2 2, 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, ... A057468
3 1 3, 7, 13, 71, 103, 541, 1091, 1367, 1627, 4177, 9011, 9551, 36913, 43063, 49681, 57917, 483611, 877843, ... A028491
3 −1 2*, 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, ... A007658
3 −2 3, 4*, 7, 11, 83, 149, 223, 599, 647, 1373, 8423, 149497, 388897, ... A057469
4 3 2, 3, 7, 17, 59, 283, 311, 383, 499, 521, 541, 599, 1193, 1993, 2671, 7547, 24019, 46301, 48121, 68597, 91283, 131497, 148663, 184463, 341233, ... A059801
4 1 2 (no others)
4 −1 2*, 3 (no others)
4 −3 3, 5, 19, 37, 173, 211, 227, 619, 977, 1237, 2437, 5741, 13463, 23929, 81223, 121271, ... A128066
5 4 3, 43, 59, 191, 223, 349, 563, 709, 743, 1663, 5471, 17707, 19609, 35449, 36697, 45259, 91493, 246497, 265007, 289937, ... A059802
5 3 13, 19, 23, 31, 47, 127, 223, 281, 2083, 5281, 7411, 7433, 19051, 27239, 35863, 70327, ... A121877
5 2 2, 5, 7, 13, 19, 37, 59, 67, 79, 307, 331, 599, 1301, 12263, 12589, 18443, 20149, 27983, ... A082182
5 1 3, 7, 11, 13, 47, 127, 149, 181, 619, 929, 3407, 10949, 13241, 13873, 16519, 201359, 396413, 1888279, ... A004061
5 −1 5, 67, 101, 103, 229, 347, 4013, 23297, 30133, 177337, 193939, 266863, 277183, 335429, ... A057171
5 −2 2*, 3, 17, 19, 47, 101, 1709, 2539, 5591, 6037, 8011, 19373, 26489, 27427, ... A082387
5 −3 2*, 3, 5, 7, 17, 19, 109, 509, 661, 709, 1231, 12889, 13043, 26723, 43963, 44789, ... A122853
5 −4 4*, 5, 7, 19, 29, 61, 137, 883, 1381, 1823, 5227, 25561, 29537, 300893, ... A128335
6 5 2, 5, 11, 13, 23, 61, 83, 421, 1039, 1511, 31237, 60413, 113177, 135647, 258413, ... A062572
6 1 2, 3, 7, 29, 71, 127, 271, 509, 1049, 6389, 6883, 10613, 19889, 79987, 608099, ... A004062
6 −1 2*, 3, 11, 31, 43, 47, 59, 107, 811, 2819, 4817, 9601, 33581, 38447, 41341, 131891, 196337, ... A057172
6 −5 3, 4*, 5, 17, 397, 409, 643, 1783, 2617, 4583, 8783, ... A128336
7 6 2, 3, 7, 29, 41, 67, 1327, 1399, 2027, 69371, 86689, 355039, ... A062573
7 5 3, 5, 7, 113, 397, 577, 7573, 14561, 58543, ... A128344
7 4 2, 5, 11, 61, 619, 2879, 2957, 24371, 69247, ... A213073
7 3 3, 7, 19, 109, 131, 607, 863, 2917, 5923, 12421, ... A128024
7 2 3, 7, 19, 79, 431, 1373, 1801, 2897, 46997, ... A215487
7 1 5, 13, 131, 149, 1699, 14221, 35201, 126037, 371669, 1264699, ... A004063
7 −1 3, 17, 23, 29, 47, 61, 1619, 18251, 106187, 201653, ... A057173
7 −2 2*, 5, 23, 73, 101, 401, 419, 457, 811, 1163, 1511, 8011, ... A125955
7 −3 3, 13, 31, 313, 3709, 7933, 14797, 30689, 38333, ... A128067
7 −4 2*, 3, 5, 19, 41, 47, 8231, 33931, 43781, 50833, 53719, 67211, ... A218373
7 −5 2*, 11, 31, 173, 271, 547, 1823, 2111, 5519, 7793, 22963, 41077, 49739, ... A128337
7 −6 3, 53, 83, 487, 743, ... A187805
8 7 7, 11, 17, 29, 31, 79, 113, 131, 139, 4357, 44029, 76213, 83663, 173687, 336419, 615997, ... A062574
8 5 2, 19, 1021, 5077, 34031, 46099, 65707, ... A128345
8 3 2, 3, 7, 19, 31, 67, 89, 9227, 43891, ... A128025
8 1 3 (no others)
8 −1 2* (no others)
8 −3 2*, 5, 163, 191, 229, 271, 733, 21059, 25237, ... A128068
8 −5 2*, 7, 19, 167, 173, 223, 281, 21647, ... A128338
8 −7 4*, 7, 13, 31, 43, 269, 353, 383, 619, 829, 877, 4957, 5711, 8317, 21739, 24029, 38299, ... A181141
9 8 2, 7, 29, 31, 67, 149, 401, 2531, 19913, 30773, 53857, 170099, ... A059803
9 7 3, 5, 7, 4703, 30113, ... A273010
9 5 3, 11, 17, 173, 839, 971, 40867, 45821, ... A128346
9 4 2 (no others)
9 2 2, 3, 5, 13, 29, 37, 1021, 1399, 2137, 4493, 5521, ... A173718
9 1 (none)
9 −1 3, 59, 223, 547, 773, 1009, 1823, 3803, 49223, 193247, 703393, ... A057175
9 −2 2*, 3, 7, 127, 283, 883, 1523, 4001, ... A125956
9 −4 2*, 3, 5, 7, 11, 17, 19, 41, 53, 109, 167, 2207, 3623, 5059, 5471, 7949, 21211, 32993, 60251, ... A211409
9 −5 3, 5, 13, 17, 43, 127, 229, 277, 6043, 11131, 11821, ... A128339
9 −7 2*, 3, 107, 197, 2843, 3571, 4451, ..., 31517, ... A301369
9 −8 3, 7, 13, 19, 307, 619, 2089, 7297, 75571, 76103, 98897, ... A187819
10 9 2, 3, 7, 11, 19, 29, 401, 709, 2531, 15787, 66949, 282493, ... A062576
10 7 2, 31, 103, 617, 10253, 10691, ... A273403
10 3 2, 3, 5, 37, 599, 38393, 51431, ... A128026
10 1 2, 19, 23, 317, 1031, 49081, 86453, 109297, 270343, ... A004023
10 −1 5, 7, 19, 31, 53, 67, 293, 641, 2137, 3011, 268207, ... A001562
10 −3 2*, 3, 19, 31, 101, 139, 167, 1097, 43151, 60703, 90499, ... A128069
10 −7 2*, 3, 5, 11, 19, 1259, 1399, 2539, 2843, 5857, 10589, ...
10 −9 4*, 7, 67, 73, 1091, 1483, 10937, ... A217095
11 10 3, 5, 19, 311, 317, 1129, 4253, 7699, 18199, 35153, 206081, ... A062577
11 9 5, 31, 271, 929, 2789, 4153, ... A273601
11 8 2, 7, 11, 17, 37, 521, 877, 2423, ... A273600
11 7 5, 19, 67, 107, 593, 757, 1801, 2243, 2383, 6043, 10181, 11383, 15629, ... A273599
11 6 2, 3, 11, 163, 191, 269, 1381, 1493, ... A273598
11 5 5, 41, 149, 229, 263, 739, 3457, 20269, 98221, ... A128347
11 4 3, 5, 11, 17, 71, 89, 827, 22307, 45893, 63521, ... A216181
11 3 3, 5, 19, 31, 367, 389, 431, 2179, 10667, 13103, 90397, ... A128027
11 2 2, 5, 11, 13, 331, 599, 18839, 23747, 24371, 29339, 32141, 67421, ... A210506
11 1 17, 19, 73, 139, 907, 1907, 2029, 4801, 5153, 10867, 20161, 293831, ... A005808
11 −1 5, 7, 179, 229, 439, 557, 6113, 223999, 327001, ... A057177
11 −2 3, 5, 17, 67, 83, 101, 1373, 6101, 12119, 61781, ... A125957
11 −3 3, 103, 271, 523, 23087, 69833, ... A128070
11 −4 2*, 7, 53, 67, 71, 443, 26497, ... A224501
11 −5 7, 11, 181, 421, 2297, 2797, 4129, 4139, 7151, 29033, ... A128340
11 −6 2*, 5, 7, 107, 383, 17359, 21929, 26393, ...
11 −7 7, 1163, 4007, 10159, ...
11 −8 2*, 3, 13, 31, 59, 131, 223, 227, 1523, ...
11 −9 2*, 3, 17, 41, 43, 59, 83, ...
11 −10 53, 421, 647, 1601, 35527, ... A185239
12 11 2, 3, 7, 89, 101, 293, 4463, 70067, ... A062578
12 7 2, 3, 7, 13, 47, 89, 139, 523, 1051, ... A273814
12 5 2, 3, 31, 41, 53, 101, 421, 1259, 4721, 45259, ... A128348
12 1 2, 3, 5, 19, 97, 109, 317, 353, 701, 9739, 14951, 37573, 46889, 769543, ... A004064
12 −1 2*, 5, 11, 109, 193, 1483, 11353, 21419, 21911, 24071, 106859, 139739, ... A057178
12 −5 2*, 3, 5, 13, 347, 977, 1091, 4861, 4967, 34679, ... A128341
12 −7 2*, 3, 7, 67, 79, 167, 953, 1493, 3389, 4871, ...
12 −11 47, 401, 509, 8609, ... A213216

*Note: if Template:Math and Template:Mvar is even, then the numbers Template:Mvar are not included in the corresponding OEIS sequence.

When Template:Math, it is Template:Math, a difference of two consecutive perfect Template:Mvarth powers, and if Template:Math is prime, then Template:Mvar must be Template:Math, because it is divisible by Template:Math.

Least Template:Mvar such that Template:Math is prime are

2, 2, 2, 3, 2, 2, 7, 2, 2, 3, 2, 17, 3, 2, 2, 5, 3, 2, 5, 2, 2, 229, 2, 3, 3, 2, 3, 3, 2, 2, 5, 3, 2, 3, 2, 2, 3, 3, 2, 7, 2, 3, 37, 2, 3, 5, 58543, 2, 3, 2, 2, 3, 2, 2, 3, 2, 5, 3, 4663, 54517, 17, 3, 2, 5, 2, 3, 3, 2, 2, 47, 61, 19, ... (sequence A058013 in the OEIS)

Least Template:Mvar such that Template:Math is prime are

1, 1, 1, 1, 5, 1, 1, 1, 5, 2, 1, 39, 6, 4, 12, 2, 2, 1, 6, 17, 46, 7, 5, 1, 25, 2, 41, 1, 12, 7, 1, 7, 327, 7, 8, 44, 26, 12, 75, 14, 51, 110, 4, 14, 49, 286, 15, 4, 39, 22, 109, 367, 22, 67, 27, 95, 80, 149, 2, 142, 3, 11, ... (sequence A222119 in the OEIS)

See alsoEdit

Template:Cols

Template:Colend

NotesEdit

Template:Notelist

ReferencesEdit

Template:Reflist

External linksEdit

Template:Sister project

|CitationClass=web }}

MathWorld linksEdit

Template:Prime number classes Template:Classes of natural numbers Template:Mersenne Template:Large numbers

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