Crack in the twin prime conjecture problem

(Image: Gemini G.E.L. (@GeminiGEL) / X)

Here is a story to convince math enthusiasts to continue to work on problems regardless of their age:

A 58-year-old lecturer who had published no original work in 12 years, and who had spent years keeping the books at a friend's Subway sandwich shop, mailed a paper to the Annals of Mathematics on April 17, 2013. Just over a month later it was accepted: pairs of consecutive primes less than 70 million apart occur infinitely often. - ScienceBlog.com

Related posts:

Prime number gaps

Twin prime sandwich

A392975 - A contribution to the On-Line Encyclopedia of Integer Sequences (OEIS)

 

(Image: Michael De Vlieger)

I recently proposed a new sequence for the On-Line Encyclopedia of Integer Sequences (OEIS). 

A Sisyphus sequence: a(0) = 0, a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest unused positive integer having the same greatest prime factor as the sum of the previous terms.

Sisyphus is a figure in Greek mythology, known as the cunning King of Corinth who was punished by Zeus to eternally roll a massive boulder up a hill in Tartarus, only for it to roll back down each time. Numerical sequences are sometimes called Sisyphus sequences when their terms climb and climb, but then drop repeatedly. 

The image above shows this process for this sequence for the first 10^6 terms. Below is a graph for how the sequence begins (first 300 terms):

One interesting feature of the sequence is the low points all appear to be prime. They begin:  7, 13, 17, 19, 23, 29, 59, 107, 137, 173, 257, 293, 467, 503...

This trend continues for at least the first 200,000 terms of the sequence.

Another feature of the sequence is that beginning with the third term, the common greatest prime factor repeats in runs of 3 or more. These factors begin:  3, 3, 3, 7, 7, 7, 7, 13, 13, 13, 13... and this trend continues for at least the first 200,000 terms of the sequence.

(Image: Sisyphus (1548–49) by Titian, Prado Museum, Madrid, Spain)

Update 4/3/2026: the sequence was published A392975 - OEIS .

Update 4/14/2026: a companion sequence, A396326, was published: Low points in A392975 (having the property of being all prime numbers).

A388038: Contribution to the OEIS (a(n) is the number of steps to reach 1 when the Sisyphus sequence (A350877) begins with n, or -1 if 1 is never reached.)

 A388038

a(n) is the number of steps to reach 1 when the Sisyphus sequence (A350877) begins with n, or -1 if 1 is never reached.

0

0, 1, 5, 2, 979, 6, 8, 3, 13, 980, 6, 7, 209, 9, 980, 4, 8, 14, 9, 981, 10, 7, 14, 8, 980, 210, 7, 10, 20, 981, 210, 5, 14, 9, 981, 15, 13, 10, 9, 982, 210, 11, 10, 8, 806, 15, 11, 9, 9, 981, 15, 211, 980, 8, 981, 11, 11, 21, 8, 982, 16, 211, 21, 6, 981, 15, 211

(list; graph; refs; listen; history; edit; text; internal format)

OFFSET

1,3

COMMENTS

Rules of the Sisyphus sequence (A350877): if the last term, s(m), is even, s(m+1) = s(m)/2; otherwise, s(m+1) = s(m) + the smallest prime not yet added.

Beginning the Sisyphus sequence with the seed 1008973 runs more than 2^30 terms without reaching 1 (A388141). Michael S. Branicky found that A388141(3584971546) = 1. - Michael De Vlieger, Sep 14 2025

LINKS

James C. McMahon, Table of n, a(n) for n = 1..10000

Michael De Vlieger, Scatterplot of first 65536 terms.

Michael De Vlieger, The Sisyphus Sequence, 2022.

EXAMPLE

For n = 1, a(n) = 0, no steps are required to reach 1.

For n = 3, a(n) = 5.

Step 1: 3 is odd, the next term is 3 + 2 (first prime) = 5.

Step 2: 5 is odd, the next term is 5 + 3 (next prime) = 8.

Step 3: 8 is even, the next term is 8/2 = 4.

Step 4: 4 is even, the next term is 4/2 = 2.

Step 5: 2 is even, the next term is 2/2 = 1.

MATHEMATICA

a[n_]:=Module[{s={n}, p=2, c=1}, While[s[[-1]]!=1, If[OddQ[s[[-1]]], AppendTo[s, s[[-1]]+p]; p=NextPrime[p], AppendTo[s, s[[-1]]/2]]; c++]; c-1]; Array[a, 67]

PROG

(Python)

from sympy import nextprime

def sisyphus(start): # generator of Sisyphus sequence beginning at start

an, p = start, 1

while True: yield an; an = an+(p:=nextprime(p)) if an&1 else an>>1

def a(n): return next(k for k, t in enumerate(sisyphus(n)) if t == 1)

print([a(n) for n in range(1, 68)]) # Michael S. Branicky, Sep 16 2025

CROSSREFS

Cf. A350877, A388141.

Sequence in context: A347426 A212481 A046270 * A139207 A137237 A369841

Adjacent sequences: A388035 A388036 A388037 * A388039 A388040 A388045

KEYWORD

look,sign,new

AUTHOR

James C. McMahon, Sep 13 2025

STATUS

approved

Important Links: https://vincico.com/seq/a350877.html

Scatterplot of terms:

Why do terms cluster near certain values (see horizontal bands in below diagram):

Also see: https://jamesmacmath.blogspot.com/2025/09/the-sisyphus-sequence-a350877.html.

A385288 - Contribution to the OEIS: Numbers with a prime number of prime factors, counted with multiplicity, and whose prime factors are each raised to a prime exponent

 A385288

Numbers with a prime number of prime factors, counted with multiplicity, and whose prime factors are each raised to a prime exponent.

0

4, 8, 9, 25, 27, 32, 49, 72, 108, 121, 125, 128, 169, 200, 243, 288, 289, 343, 361, 392, 500, 529, 675, 800, 841, 961, 968, 972, 1125, 1323, 1331, 1352, 1369, 1372, 1568, 1681, 1800, 1849, 2048, 2187, 2197, 2209, 2312, 2700, 2809, 2888, 3087, 3125, 3267, 3481

(list; graph; refs; listen; history; edit; text; internal format)

OFFSET

1,1

COMMENTS

a(n) = A114129(n) through n=25; then a(26) = 961 and A114129(26) = 864.

Subset of A056166.

Subset of A001694. - Michael De Vlieger, Jun 25 2025.

LINKS

James C. McMahon, Table of n, a(n) for n = 1..1000

EXAMPLE

200 = 2^3 * 5^2; 200 has a prime number of prime factors, counted with multiplicity (3 + 2 = 5), and exponents 3 and 2 are prime.

MATHEMATICA

Select[Range[10^4], AllTrue[Last/@FactorInteger[#], PrimeQ]&&PrimeQ[PrimeOmega[#]]&]

PROG

(PARI) isok(k) = my(f=factor(k)); isprime(bigomega(k)) && (sum(k=1, #f~, isprime(f[k, 2])) == omega(f)); \\ Michel Marcus, Jun 25 2025

CROSSREFS

Cf. A001248, A001694, A030078, A056166, A114129.

Sequence in context: A383211 A051676 A114129 * A383103 A053810 A076702

Adjacent sequences: A385283 A385284 A385286 * A385289 A385292 A385293

KEYWORD

nonn,new

AUTHOR

James C. McMahon, Jun 24 2025

STATUS

approved

Online Encyclopedia of Integer Sequences, OEIS.

A380943 Contribution to the OEIS (Saint Patrick Primes)

 A380943

Primes written in decimal representation by the concatenation of primes p and q such that the concatenation of q and p also forms a prime.

+40
0

37, 73, 113, 173, 197, 311, 313, 317, 331, 337, 359, 367, 373, 593, 617, 673, 719, 733, 761, 797, 977, 1093, 1097, 1123, 1277, 1319, 1361, 1373, 1783, 1913, 1931, 1979, 1997, 2293, 2297, 2311, 2347, 2389, 2713, 2837, 2971, 3109, 3119, 3137, 3191, 3229, 3271

(list; graph; refs; listen; history; edit; text; internal format)

OFFSET

1,1

COMMENTS

Member primes could be named Saint Patrick primes because the date of Saint Patrick's Day, March 17 (3/17), produces the terms 173 and 317.

LINKS

James C. McMahon, Table of n, a(n) for n = 1..1000

Michael De Vlieger, Plot a(n) = p<>q at (x,y) = (pi(p), pi(q)) showing all terms with pi(p) <= 40 and pi(q) <= 40, labeling p in red and q in blue.

Michael De Vlieger, Plot a(n) = p<>q at (x,y) = (pi(p), pi(q)) showing all terms with pi(p) <= 2000 and pi(q) <= 2000.

EXAMPLE

Primes 173 and 317 are members because they are formed by the concatenation of 17 & 3 and 3 & 17, respectively.

While the concatenation of 13 and 7 forms the prime 137, it is not a member because the concatenation of 7 and 13 is 713, which is not prime.

MATHEMATICA

lim=3300; plim=Max[FromDigits[Rest[IntegerDigits[lim]]], FromDigits[Drop[IntegerDigits[lim], -1]]]; p=Prime[Range[PrimePi[plim]]]; p2=Subsets[p, {2}]; fp[{a_, b_}]:=FromDigits[Join[IntegerDigits[a], IntegerDigits[b]]]; rfp[{a_, b_}]:=FromDigits[Join[IntegerDigits[b], IntegerDigits[a]]]; pabba=Select[p2, PrimeQ[fp[#]]&&PrimeQ[rfp[#]]&]; pp1=fp/@pabba; pp2=rfp/@pabba; Select[Union[Join[pp1, pp2]], #<=lim&]

PROG

(Python)

from sympy import isprime

def ok(n):

if not isprime(n): return False

s = str(n)

return any(s[i]!='0' and isprime(int(s[:i])) and isprime(int(s[i:])) and isprime(int(s[i:]+s[:i])) for i in range(1, len(s)))

print([k for k in range(3300) if ok(k)]) # Michael S. Branicky, Apr 05 2025

CROSSREFS

Subsequence of A019549 and A105184.

Cf. A133187.

KEYWORD

nonn,base,new

AUTHOR

James C. McMahon, Apr 03 2025

STATUS

approved

See post: https://jamesmacmath.blogspot.com/2025/03/saint-patrick-another-prime-day.html

Also, Michael De Vlieger made interesting graphics for this sequence: https://oeis.org/A380943/a380943.png and https://oeis.org/A380943/a380943_1.png

Saint Patrick - Another Prime Day

(Image: By Internet Archive Book Images -  https://archive.org/stream/irelandscrownoft00ryan/irelandscrownoft00ryan#page/n61/mode/1up, No restrictions, https://commons.wikimedia.org/w/index.php?curid=41964622)

A recent post about Julius Caesar introduced the concept of Caesar Primes. Caesar's birthday is the 13th of July (the 7th month), and 137 is a prime number. Caesar Primes are primes formed by the concatenation of q and p, where q > p are also primes.

As we celebrate Saint Patrick's Day on March 17, we note that 317 and 173 are prime numbers. This suggests a sequence of primes similar to the Caesar Primes. Other pairs of primes formed by two primes include: {37, 73}, {113, 311}, {313,331},{337,733}, {359,593}...

Update 4/8/2025. I submitted a proposed sequence to the On-Line Encyclopedia of Integer Sequences (OEIS). It is currently under review as https://oeis.org/draft/A380943, Primes written in decimal representation by the concatenation of primes p and q such that the concatenation of q and p also forms a prime.

The sequence begins: 37, 73, 113, 173, 197, 311, 313, 317, 331, 337, 359, 367, 373, 593, 617, 673, 719, 733, 761, 797, 977, 1093, 1097, 1123, 1277, 1319, 1361, 1373, 1783, 1913, 1931, 1979, 1997, 2293, 2297, 2311, 2347, 2389, 2713, 2837, 2971, 3109, 3119, 3137, 3191, 3229, 3271...

Update 4/13/2025 - the proposed sequence was published in the OEIS: https://oeis.org/A380943.

Julius Caesar and Caesar Primes

 

(Image:https://commons.wikimedia.org/wiki/File:Retrato_de_Julio_C%C3%A9sar_(26724093101).jpg)

March 15 is also known as the Ides of March, when Julius Caesar was assassinated in 44 BCE. Caesar was born on July 12 or 13, 100 BCE (records vary on the date of July 12 or July 13). For the purpose of this post, July 13 is consistent with the numbers known as Caesar Primes. One such number is 137 - the concatenation of two primes, 13 and 7 (with Caesar's birthday being the 13th day of the 7th month).

The list of Caesar Primes is found in the On-Line Encyclopedia of Integer Sequences (OEIS) as sequence: https://oeis.org/A133187. The sequence consists of the prime numbers formed by the concatenation of q and p, where q > p are also primes.

I wrote a Mathematica  to produce this sequence:

lim=2700;plim=Max[FromDigits[Rest[IntegerDigits[lim]]],FromDigits[Drop[IntegerDigits[lim],-1]]];f2p[{p_,q_}]:=FromDigits[Join[IntegerDigits[q],IntegerDigits[p]]];p=Prime[Range[PrimePi[plim]]];p2=Subsets[p,{2}];Union[Select[f2p/@p2,PrimeQ[#]&&#<=lim&]]

The first 46 Caesar Primes are: 53,  73, 113, 137, 173, 193, 197, 233, 293, 313, 317, 373, 433, 593, 613, 617, 673, 677, 733, 797, 977, 1013, 1033, 1093, 1097, 1277, 1373, 1493, 1637, 1733, 1913, 1933, 1973, 1993, 1997, 2113, 2237, 2273, 2293, 2297, 2311, 2333, 2393, 2417, 2633, 2693.

A381149 Contribution to the OEIS

 A381149

a(1) = 2, a(2) = 3; thereafter, a(n) = a(n-1) + sum of prior prime terms.

+40
0

2, 3, 8, 13, 31, 80, 129, 178, 227, 503, 1282, 2061, 2840, 3619, 4398, 5177, 5956, 6735, 7514, 8293, 17365, 26437, 61946, 97455, 132964, 168473, 203982, 239491, 275000, 310509, 346018, 381527, 798563, 1215599, 1632635, 2049671, 2466707, 5350450, 8234193, 11117936

(list; graph; refs; listen; history; edit; text; internal format)

OFFSET

1,1

LINKS

James C. McMahon, Table of n, a(n) for n = 1..10000

EXAMPLE

For n=5, a(5) = a(4) + sum of prior primes = 13 + (2 + 3 + 13) = 31, so that 13 is counted twice.

MATHEMATICA

Nest[Append[#, #[[-1]]+Total[Select[#, PrimeQ]]]&, {2, 3}, 38]

PROG

(Python)

from sympy import isprime

from itertools import count, islice

def agen(): # generator of terms

yield from [2, 3]

primesum, an = 5, 3

while True:

an += primesum

if isprime(an): primesum += an

yield an

print(list(islice(agen(), 40))) # Michael S. Branicky, Feb 19 2025

CROSSREFS

Cf. A101135, A119746, A131093, A381150.

KEYWORD

nonn,new

AUTHOR

James C. McMahon, Feb 15 2025

STATUS

approved

This sequence is similar to my other sequence submitted at the same time, A381150. However, I think A381150 is more interesting because of the terms that repeat and also how it cycles between positive and negative terms. Link to prior post: https://jamesmacmath.blogspot.com/2025/02/a-sequence-with-competing-sums-of-prior.html

A381150 Contribution to the OEIS

   

Recently proposed for the On-Line Encyclopedia of Integer Sequences (OEIS) is the sequence:

a(1) = 2, a(2) = 3; thereafter, a(n) = a(n-1) + (sum of prior prime terms or whose negatives are prime) - (sum of prior composite terms or whose negatives are composite).

   The sequence starts: 2, 3, 8, 5, 7, 16, 9, -7, -30, -23, -39, -16, 23, 85, 62, -23, -131... and continues to cycle through negative and positive integers. Although the sequence is not periodic, it does repeat terms. For example, the  term 347 (or its negative) occurs nine times in the first 48 terms. The 418th (or its negative) occurs 71 times in a span of 211 terms.

    My colleague, Michael De Vlieger, produced the graphic showing the sequence's progression shown above.

    If approved, the sequence will become A381150. Currently, it is still in draft form: https://oeis.org/A381150.

    Update: The sequence was published 2/25/2025.

A381150

a(0) = 1, a(1) = 2, a(2) = 3; thereafter, a(n) = a(n-1) + (sum of prior prime terms or whose negatives are prime) - (sum of prior composite terms or whose negatives are composite).

0

1, 2, 3, 8, 5, 7, 16, 9, -7, -30, -23, -39, -16, 23, 85, 62, -23, -131, -370, -239, -347, -802, -455, 347, 1496, 1149, -347, -2190, -1843, 347, 2884, 2537, -347, -3578, -3231, 347, 4272, 3925, -347, -4966, -4619, 347, 5660, 5313, -347, -6354, -6007, -11667, -5660

(list; graph; refs; listen; history; edit; text; internal format)

OFFSET

0,2

LINKS

Michael De Vlieger, Table of n, a(n) for n = 0..10003 (a(n) for n = 1..1000 from James C. McMahon)

Michael De Vlieger, Scatterplot of m*log_10(m*a(n)), n = 1..2^10, where m = 1 of a(n) > 0 (shown in green) and m = -1 if a(n) < 0 (shown in red).

Michael De Vlieger, Scatterplot of m*log_10(m*a(n)), n = 1..2^16, where m = 1 of a(n) > 0 (shown in green) and m = -1 if a(n) < 0 (shown in red).

EXAMPLE

For n=5, a(5) = 5 + (2 + 3 + 5) - 8 = 7.

For n=9, a(9) = -7 + (2 + 3 + 5 + 7 -7) - (8 + 16 + 9) = -7 + 10 - 33 = -30

MAPLE

b:= proc(n) option remember; `if`(n<1, 0, b(n-1)+(t->

`if`(isprime(abs(t)), t, `if`(abs(t)>1, -t, 0)))(a(n)))

end:

a:= proc(n) option remember; `if`(n<3, n+1, a(n-1)+b(n-1)) end:

seq(a(n), n=0..48); # Alois P. Heinz, Feb 15 2025

MATHEMATICA

Nest[Append[#, #[[-1]]+Total[Select[#, PrimeQ]]-Total[Select[#, CompositeQ]]]&, {1, 2, 3}, 46]

CROSSREFS

Cf. A000040, A002808, A101135, A119746, A131093, A381149.

Sequence in context: A284047 A265343 A347193 * A268392 A154760 A231817

Adjacent sequences: A381147 A381148 A381149 * A381152 A381153 A381154

KEYWORD

sign,new

AUTHOR

James C. McMahon, Feb 15 2025

STATUS

approved

A380198 Contribution to the OEIS

A380198       Difference between pi(2^n) and the integer nearest to 2^n / log(2^n).

0

-2, -1, 0, 0, 2, 3, 5, 8, 15, 24, 40, 72, 119, 212, 360, 633, 1128, 1989, 3580, 6386, 11537, 20897, 37980, 69354, 127336, 234054, 431877, 799754, 1484440, 2763961, 5156791, 9644970, 18080775, 33959344, 63902732, 120474951, 227515953, 430345298, 815241632

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Michael De Vlieger, Table of n, a(n) for n = 1..92

FORMULA

a(n) = - A053622(2^n).

a(n) = A007053(n) - A050499(2^n).

EXAMPLE

n 2^n pi(2^n) round(2^n/log(2^n) a(n)

------------------------------------------------

1 2 1 3 -2

2 4 2 3 -1

3 8 4 4 0

4 16 6 6 0

MATHEMATICA

Table[PrimePi[2^n]-Round[2^n/Log[2^n]], {n, 39}]

CROSSREFS

Cf. A000720, A007053, A050499, A053622, A057835.

Sequence in context: A378340 A063181 A247310 * A059220 A261630 A301503

Adjacent sequences: A380189 A380190 A380197 * A380208 A380209 A380210

KEYWORD

sign,new

AUTHOR

James C. McMahon, Jan 16 2025

STATUS

approved

Very Large and Easy-to-Remember Prime Numbers

There was a recent article about asking people to name a prime number. One answer was given as 2^31-1; however, the responder was asked to recite the digits. He couldn't remember the digits of 2^31-1, which are 2,147,483,647. His friend, Neil  Sloane, offered another large prime: 12,345,678,910,987,654,321. It is easy to remember because one just needs to count from 1 to 10 and then back down to 1. The original story is recounted in this Scientific American article:  These Prime Numbers Are So Memorable That People Hunt for Them.

The number cited above is found in the OEIS sequence A350153.  

Other related sequences are:

A048847, Primes formed by concatenation of first k odd numbers.

A173426, a(n) is obtained by starting with 1, sequentially concatenating all decimal numbers up to n, and then, starting from n-1, sequentially concatenating all decimal numbers down to 1.

A260343, Numbers n such that the base-n number formed by concatenating the base-n numbers 1 2 ... n-1 n n-1 ... 2 1 is prime.

A323532, Numbers k such that the decimal concatenation of the numbers from 1 up to k followed by digit reversals of the numbers from (k-1) down to 1 is a prime.

Primes that can be expressed in the form p² + 4q², where both p and q are prime numbers

In a significant advancement for number theory, mathematicians Ben Green of the University of Oxford and Mehtaab Sawhney of Columbia University have introduced a novel method for identifying specific types of prime numbers. Their work, detailed in a recent Quanta Magazine article, focuses on primes that can be expressed in the form p² + 4q², where both p and q are prime numbers.

Quanta Magazine

Prime numbers, defined as numbers greater than 1 that have no positive divisors other than 1 and themselves, are fundamental to mathematics. Understanding their distribution has been a longstanding challenge. While the infinitude of primes was established by Euclid around 300 BCE, identifying primes that satisfy additional constraints has proven difficult. Green and Sawhney's achievement in proving the existence of infinitely many primes of the form p² + 4q² represents a significant breakthrough in this area.

Their approach diverged from traditional methods by incorporating tools from other mathematical disciplines, demonstrating the potential for interdisciplinary techniques to address complex problems in number theory. This innovative strategy not only resolved a specific conjecture but also opened avenues for applying similar methods to other mathematical challenges.

The implications of this discovery extend beyond the immediate result. By enhancing our understanding of prime distribution, it contributes to the broader field of analytic number theory and may influence related areas such as cryptography, where prime numbers play a crucial role.

For a more comprehensive exploration of Green and Sawhney's work and its significance, the full article is available on Quanta Magazine's website.

Quanta Magazine

New Record-Breaking Prime Number Discovered, 2^136,279,841-1

 

New Record-Breaking Prime Number Discovered

Mathematicians have discovered a new record-breaking prime number, the largest known prime to date. The number, 2^136,279,841-1, is a Mersenne prime, a type of prime number that is one less than a power of two.

The discovery was made by the Great Internet Mersenne Prime Search (GIMPS), a distributed computing project that uses volunteers' computers to search for Mersenne primes. The new prime was found by a volunteer in the United States.

The previous record for the largest known prime number was held by 2^77,232,917-1, which was discovered in 2018. The new prime is more than twice as large as the previous record holder.

The discovery of new prime numbers is important for several reasons. First, they are used in cryptography, which is the science of secure communication. Second, they are used in number theory, a branch of mathematics that studies the properties of numbers. Third, they are simply interesting in their own right.

The discovery of the new prime number is a major milestone for the GIMPS project. It is also a testament to the power of distributed computing. By harnessing the power of thousands of volunteers' computers, GIMPS has been able to make significant contributions to the field of mathematics.

What is a Mersenne Prime?

A Mersenne prime is a prime number that is one less than a power of two. In other words, it is a number of the form 2^n - 1, where n is a positive integer.

Mersenne primes are named after Marin Mersenne, a French monk who lived from 1588 to 1648. Mersenne studied these numbers and made a list of all the Mersenne primes up to 2^257 - 1. However, his list was not entirely correct.

Since Mersenne's time, many more Mersenne primes have been discovered. The largest known Mersenne prime is 2136,279,841-1.

Why are Mersenne Primes Important?

Mersenne primes are important for several reasons. First, they are used in cryptography. The RSA cryptosystem, which is one of the most widely used public-key cryptosystems, uses large prime numbers, including Mersenne primes.

Second, Mersenne primes are used in number theory. Number theorists study the properties of Mersenne primes and other types of prime numbers.

Third, Mersenne primes are simply interesting in their own right. They are a fascinating example of a mathematical pattern that has been studied for centuries.

Great Internet Mersenne Prime Search (GIMPS)

The Great Internet Mersenne Prime Search (GIMPS) is a distributed computing project that uses volunteers' computers to search for Mersenne primes. The project was founded in 1996 by George Woltman.

GIMPS has discovered several Mersenne primes, including the largest known Mersenne prime, 2136,279,841-1. The project is a great way for people to contribute to mathematics and science.

If you are interested in learning more about Mersenne primes or the GIMPS project, I encourage you to visit the GIMPS website at https://www.mersenne.org/.

News about this recently discovered prime: 

https://www.livescience.com/physics-mathematics/mathematics/largest-known-prime-number-spanning-41-million-digits-discovered-by-amateur-mathematician-using-free-software

https://www.newscientist.com/article/2452686-amateur-sleuth-finds-largest-known-prime-number-with-41-million-digits/

Numberphile video: https://www.numberphile.com/videos/man-who-found-the-worlds-biggest-prime

Determining if a very large number is Prime: 

https://www.scientificamerican.com/article/how-to-identify-a-prime-number-without-a-computer/