Marin Mersenne

How to use this course

Marin Mersenne is remembered in three places. Physicists meet him in the laws that give the frequency of a stretched string. Number theorists meet him in the primes \(2^p-1\), the largest primes known today. Historians meet him as the man at the centre of European science between about 1620 and 1648, the friar who copied, forwarded and provoked the letters of Descartes, Fermat, Galileo, Gassendi, Roberval, Hobbes and the Pascals.

This course treats all three. Part I follows his formation as a Catholic priest and scholar. Part II works through the science: four lessons on sound and motion, where Mersenne was an experimenter, and two on number theory, where his short list of exponents set a problem that took three centuries to settle. Part III returns to the correspondence network and his legacy.

Each technical lesson opens with an italic paragraph for the general reader, then gives precise statements, derivations and exercises with folded solutions. Modern notation is used throughout. Mersenne himself wrote in words and proportions, with no algebra beyond what a geometer of 1630 would use.

Timeline

Maine, Le Mans, La Flèche, 1588–1611

Marin Mersenne was born on 8 September 1588 near the village of Oizé, in the old province of Maine, south of Le Mans. His parents, Julien Mersenne and Jeanne Moulière, were country people of modest means. He was baptised the day he was born. France was then in the last years of the Wars of Religion, and the boy grew up in a kingdom where questions of faith were questions of public order. That fact shaped his whole career.

He was pious and studious, and his parents sent him to the Collège du Mans for grammar. In 1604, aged about sixteen, he entered the Jesuit college that Henri IV had just founded at La Flèche. René Descartes, eight years younger, arrived there a few years later. The two did not become close at school; their friendship belongs to the 1620s. What La Flèche gave both of them was the Jesuit curriculum: Aristotle in the scholastic manner, but also a serious course in mathematics drawn from Christopher Clavius, in which arithmetic, geometry, astronomy and music were treated as one family of mathematical sciences.

Paris

From La Flèche he moved to Paris, attending philosophy lectures at the Collège Royal and theology at the Sorbonne, and adding Greek and Hebrew. He considered the secular priesthood. Instead, in July 1611, he took the habit of the Minims, a strict mendicant order founded in the fifteenth century by Saint Francis of Paola. Minims kept a perpetual Lenten diet and a rule of humility; their name means “the least”. After a novitiate at Nigeon, near Paris, and a period at Meaux, he was ordained priest within about two years.

The choice matters for everything that follows. A Minim owned nothing and moved where his superiors sent him. In return, the order gave Mersenne a cell, a library, freedom from the need to earn a living, and the leisure for a life of study that a provincial peasant’s son could not otherwise have had.

The Minim friar, 1611–1623

From 1614 to 1618 Mersenne taught philosophy and theology to young members of the order at the Minim house in Nevers. One of his pupils, Hilarion de Coste, became his lifelong confidant and, in 1649, his first biographer. Around 1619 or 1620 Mersenne was recalled to Paris, to the Minim convent of the Annonciade near the Place Royale, the square now called the Place des Vosges. He lived there for the rest of his life, apart from journeys.

The Place Royale was at the heart of aristocratic Paris, a short walk from the Marais townhouses of magistrates, financiers and scholars. A friar there could receive visitors, borrow books and send letters through the royal post. Mersenne’s cell became, over the next two decades, a reading room, a laboratory for simple experiments with strings and pendulums, and a sorting office for mail from all over Europe.

A scholar of the Counter-Reformation

Mersenne’s first ambitions were theological. Paris in the early 1620s was alarmed by libertines, deists and sceptics, and by a fashion for Renaissance natural magic, cabbala and the Rosicrucian manifestos that had appeared in Germany in 1614–15. Mersenne set out to show that true natural knowledge supports religion, and that magic and occult philosophy corrupt both. He thought the best weapon against the occult was the new mathematical and experimental science.

From apologetics to natural philosophy, 1623–1630

In 1623 Mersenne published Quaestiones celeberrimae in Genesim, a huge folio. It is nominally a commentary on the first chapters of Genesis, and in practice an encyclopedia aimed at magicians, atheists and deists, with long digressions on music, optics and mathematics. In the same year he published two shorter books on reason and the spiritual life. The Quaestiones brought him into contact with Pierre Gassendi, who became his closest friend.

Two French books followed: L’Impiété des déistes (1624) and La Vérité des sciences contre les sceptiques ou Pyrrhoniens (1625). The second is a dialogue in which a Christian philosopher answers a sceptic. Mersenne concedes that we cannot know the inner essences of things. He insists that we can still know their measurable effects, and that mathematics and careful experiment give knowledge certain enough to live by. Historians call this position mitigated scepticism. Richard Popkin read it as an early statement of the modern scientific attitude; others, including Peter Dear and Daniel Garber, place Mersenne closer to traditional scholastic thought. Either way, it gave him a philosophy that welcomed experiment without demanding a complete system.

The Fludd controversy

Mersenne attacked the English physician Robert Fludd, a defender of the Rosicrucians and of a cosmic, harmonic natural magic. Fludd replied with scorn, and at Mersenne’s request Gassendi wrote a reply that appeared in 1630. Johannes Kepler had already crossed swords with Fludd. The quarrel shows Mersenne drawing the line he cared about: music and the cosmos could be studied mathematically, but not through hidden sympathies.

Turning to music and mechanics

In 1626 he issued an edition of Euclid and a Synopsis mathematica of classical texts, and in 1627, under the anagrammatic pseudonym Sieur de Sermes, a Traité de l’harmonie universelle. Music was the natural bridge. In the Jesuit scheme it was a mathematical science, the study of numbers in sound. Mersenne began to ask what physical motions those numbers measured. A journey to the Netherlands in 1629–30, partly to take the waters for his health, introduced him to Isaac Beeckman, the Dutch schoolmaster who had already speculated that pitch is the frequency of vibration, and renewed his acquaintance with Descartes, who had settled in Holland. From then on Mersenne was Descartes’s main correspondent and agent in Paris.

The laws of the vibrating string

For the general reader. Pluck a guitar string and the note depends on three things: how long the string is, how tightly it is stretched, and how heavy it is. A shorter, tighter or lighter string sounds higher. Mersenne turned this everyday knowledge into exact proportions by measuring, and in doing so made the pitch of a note into a physical number, the frequency of vibration.

The background

Pythagorean tradition knew that halving the length of a string raises its pitch by an octave, and that lengths in ratio \(3:2\) and \(4:3\) give the fifth and fourth. Vincenzo Galilei, father of Galileo, had found in the 1580s that the same intervals need very different ratios when produced by changing tension: to raise a string an octave by hanging weights from it, one must quadruple the weight. In the 1610s and 1620s Beeckman, Descartes and Galileo each argued that pitch corresponds to the number of vibrations per unit time. Mersenne’s contribution was a systematic set of measurements on strings of brass, steel, gut and other materials, with weights, lengths and diameters varied one at a time. He set out the results in Harmonie universelle (1636). Galileo’s Two New Sciences (1638) contains essentially the same relations; the laws are rightly called Mersenne’s because he tested them most fully and published them first.

Derivation from the wave equation

Mersenne had the proportions; the dynamics came a century later from Brook Taylor (1713), Jean le Rond d’Alembert (1747) and Leonhard Euler. Let \(y(x,t)\) be the small transverse displacement of a string under tension \(T\), with \(0\le x\le L\). A short piece of length \(dx\) has mass \(\mu\,dx\). The vertical component of tension at each end is \(T\,\partial y/\partial x\) to first order in the slope, so Newton’s second law gives

With \(y(0,t)=y(L,t)=0\), separation of variables gives the standing waves

Real strings have some bending stiffness, which raises the higher modes slightly above \(nf_1\), and stretching a string to increase its tension also lengthens and thins it. These effects help explain why Mersenne’s measurements were good but not exact.

Consonance, ratios and temperament

For the general reader. Why do some pairs of notes sound smooth together and others rough? The ancient answer was that pleasant intervals correspond to simple ratios such as 2:1 or 3:2. Once Mersenne and his contemporaries identified pitch with frequency, the old ratios became ratios of vibrations, and a physical explanation became possible. The same arithmetic shows why no keyboard can be perfectly in tune in every key.

Intervals as ratios

An interval between frequencies \(f_1 \lt f_2\) is the ratio \(r=f_2/f_1\). Intervals add when ratios multiply, so a logarithmic unit is natural. The cent divides the octave into 1200 equal parts:

The octave is \(2:1\) (1200 cents), the fifth \(3:2\) (about 702 cents), the fourth \(4:3\) (about 498), the major third \(5:4\) (about 386).

The coincidence theory

Mersenne, following Beeckman, explained consonance by the coincidence of vibrations. If two strings vibrate in ratio \(3:2\), every third vibration of the higher string coincides with every second vibration of the lower, and the air receives a regular pattern of joint pulses. The simpler the ratio, the more often the coincidences and the smoother the sound. He also tested the Pythagorean prediction for tension directly; by his report, the ear judged an octave at a weight ratio somewhat above the theoretical \(4:1\), nearer \(4\tfrac14:1\). Modern acoustics refines the coincidence picture through the overtone series of Lesson 06: two tones blend when their upper partials coincide rather than beat.

Proof. The equation means \(3^m=2^{m+n}\). The left side is odd and the right side even for \(m+n\ge1\). By unique factorization, this is impossible. \(\square\)

Twelve fifths come close to seven octaves. The excess

is the Pythagorean comma. Every tuning system must distribute it somehow.

Equal temperament

Equal temperament spreads the comma evenly, making every semitone the ratio \(2^{1/12}\approx1.059463\), exactly 100 cents. The number \(2^{1/12}\) is irrational, and lute makers had long used rational approximations: Vincenzo Galilei recommended \(18:17\), about 99 cents. In Harmonie universelle Mersenne discussed equal temperament at length, printed tables of string lengths for it computed by his correspondents Jean Beaugrand, Ismaël Boulliau and Jean Galle, and gave a geometric construction for the semitone that is accurate to well under a cent. He was among the first writers to treat equal temperament as a serious option for all instruments.

Counting vibrations; overtones

For the general reader. A musical string vibrates hundreds of times a second, far too fast to count by eye. Mersenne found a way around the problem: use his laws on a very long, slack string that vibrates slowly enough to count, then scale up. He also noticed something puzzling: a single plucked string sounds several pitches at once.

An absolute frequency

Mersenne’s laws are statements of proportion. They say how frequency changes, but not what it is. To fix a number, Mersenne stretched long cords whose slow swings could be counted against a clock or pulse, then used the inverse-length law to infer the frequency of a shorter string of the same material and tension. The result in Harmonie universelle is usually described as the first absolute determination of the frequency of an audible tone; he put a low note at about 84 vibrations per second. The method is exact in principle, since it uses only the proportionality \(f\propto1/L\).

Harmonics

Listening closely to single strings, Mersenne reported that besides the main note he could hear fainter higher tones, among them the twelfth and the major seventeenth. He could not explain how one string could sound several pitches simultaneously. The explanation emerged in the eighteenth century, through Joseph Sauveur, who named the harmonics and the nodes, and Daniel Bernoulli, who proposed that a string’s motion is a superposition of its modes. Lesson 04 gives the modes: a general motion is

The \(n\)th mode sounds at \(nf_1\): the octave (\(n=2\)), twelfth (\(n=3\)), double octave (\(n=4\)) and major seventeenth (\(n=5\), two octaves and a major third). These are exactly the tones Mersenne heard.

The amplitudes fall like \(1/n^2\), so the low harmonics dominate, and every mode with a node at the plucking point is missing.

The speed of sound

For the general reader. At a distance, you see the flash of a gun before you hear the bang. Mersenne used that delay to measure how fast sound travels. His answer was too high by about a third, but the method was sound, and the question of what sets the speed of sound led to some of the best physics of the next two centuries.

Mersenne’s measurements

In Harmonie universelle Mersenne reported timing the interval between the flash of a firearm and the arrival of its report over a measured distance. He obtained about 1,380 Paris feet per second. A second method, timing an echo from a wall at known distance, gave a much lower figure, about 970 Paris feet per second; the discrepancy led some to suppose that reflected sound travels more slowly. Gassendi, with a similar flash method, found 1,473 Paris feet per second.

Theory

Isaac Newton gave the first theoretical value in the Principia (1687), treating sound as an isothermal compression wave: \(c=\sqrt{p/\rho}\). This gives about 280 m/s, too low. Pierre-Simon Laplace (1816) corrected the formula by noting that compressions are adiabatic:

with \(\gamma\approx1.40\) for air and \(\theta\) the temperature in degrees Celsius. At 15 °C this is about 340 m/s. The speed is independent of pressure and, in the audible range, of frequency.

Falling bodies and the pendulum

For the general reader. Galileo claimed that a falling body covers distances proportional to the square of the time, and that a pendulum’s swings take the same time whether large or small. Mersenne set out to test both claims. He confirmed the first approximately, and found that the second is not quite true, a correction that later led Christiaan Huygens to the cycloidal pendulum.

Galileo in French

In the early 1620s Mersenne still listed Galileo among innovators to be resisted. By the early 1630s he was one of his most active supporters outside Italy. In 1634 he published Les Méchaniques de Galilée, a French version of Galileo’s unpublished treatise on simple machines, and in 1639 Les Nouvelles Pensées de Galilée, a paraphrase of the Two New Sciences. In 1634 he reported his own falling-body trials from heights of 48, 108 and 147 feet. They broadly supported the time-squared law, though Mersenne was candid about how hard such timing was.

The pendulum

A simple pendulum of length \(\ell\) obeys \(\ddot\theta=-(g/\ell)\sin\theta\). For small swings \(\sin\theta\approx\theta\), giving simple harmonic motion with period \(T_0=2\pi\sqrt{\ell/g}\), independent of amplitude. Mersenne is credited with the first measurement of the length of the seconds pendulum, whose single swing lasts one second (\(T_0=2\) s), and with the first observation that large swings take measurably longer than small ones, contrary to Galileo.

Derivation. Energy conservation gives \(\dot\theta^2=\frac{2g}{\ell}(\cos\theta-\cos\theta_0)=\frac{4g}{\ell}\big(k^2-\sin^2\tfrac\theta2\big)\). Substitute \(\sin\tfrac\theta2=k\sin\phi\); a quarter-period runs from \(\phi=0\) to \(\pi/2\) and the integral above follows. Expanding \((1-k^2\sin^2\phi)^{-1/2}=1+\tfrac12k^2\sin^2\phi+\cdots\) and using \(\int_0^{\pi/2}\sin^2\phi\,d\phi=\pi/4\) gives \(T=T_0(1+\tfrac14k^2+\cdots)\), and \(k^2\approx\theta_0^2/4\).

Numbers of the form \(2^p-1\)

For the general reader. Double 1 repeatedly and subtract 1: 1, 3, 7, 15, 31, 63, 127, … Some of these numbers are prime. Mersenne published a bold claim about exactly which ones are prime up to an enormous size. He was wrong in five places, but his claim fixed attention on these numbers for good, and they now carry his name.

Write \(M_n=2^n-1\). The interest in these numbers is ancient: Euclid proved that \(2^{p-1}M_p\) is a perfect number (equal to the sum of its proper divisors) whenever \(M_p\) is prime. In the 1630s and 1640s perfect numbers were a favourite topic in Mersenne’s circle, and he passed problems about them between Frenicle de Bessy, Fermat and others.

Proof. If \(n=ab\) with \(1 \lt a \lt n\), then \(2^{ab}-1=(2^a-1)\big(1+2^a+2^{2a}+\cdots+2^{(b-1)a}\big)\), and \(1 \lt 2^a-1 \lt M_n\). \(\square\) The converse is false: \(M_{11}=2047=23\cdot89\).

Proof. \(q\mid2^p-1\) means \(2^p\equiv1\pmod q\), so the order of 2 modulo \(q\) divides \(p\). It is not 1, so it equals \(p\). By Fermat’s little theorem \(2^{q-1}\equiv1\), so \(p\mid q-1\). As \(q\) and \(p\) are odd, \(q-1\) is even and divisible by \(p\), hence by \(2p\). For the second claim, \(2^{(q-1)/2}=(2^{p})^{k}\equiv1\pmod q\), so 2 is a quadratic residue mod \(q\), and by the second supplement to quadratic reciprocity \(q\equiv\pm1\pmod8\). \(\square\)

Fermat stated the first part in a letter of June 1640, answering a challenge from Frenicle relayed by Mersenne, and used it to show that \(M_{37}=137\,438\,953\,471\) is divisible by \(223=2\cdot3\cdot37+1\). Only the second candidate needed testing: 149 fails, 223 divides.

Proof of the first part. The divisor-sum function \(\sigma\) is multiplicative, so \(\sigma(N)=\sigma(2^{p-1})\sigma(M_p)=(2^p-1)\cdot2^p=2N\). Euler’s converse, published posthumously, writes \(N=2^{a}m\) with \(m\) odd and shows from \(\sigma(N)=2N\) that \(m\) must be a prime of the form \(2^{a+1}-1\).

The 1644 claim

In the preface to Cogitata physico-mathematica (1644), Mersenne asserted that among primes \(p\le257\), \(M_p\) is prime exactly for

No one in 1644 could test numbers of this size. Stillman Drake suggested Frenicle as a possible source and proposed that some entries may be misprints. The final scorecard:

• \(M_{31}\) was proved prime by Euler in 1772.
• \(M_{127}\) was proved prime by Édouard Lucas in 1876, who also showed \(M_{67}\) composite without finding a factor.
• \(M_{61}\), omitted by Mersenne, was proved prime by Ivan Pervushin in 1883.
• In 1903 Frank Nelson Cole factored \(M_{67}=193\,707\,721\times761\,838\,257\,287\).
• \(M_{89}\) (1911) and \(M_{107}\) (1914), both omitted, were found prime by R. E. Powers.
• \(M_{257}\) was eventually shown composite; with that, every case up to 257 was settled, and Mersenne’s list contained exactly five errors.

The Lucas–Lehmer test and the modern search

For the general reader. Testing whether a number with millions of digits is prime sounds hopeless. For Mersenne numbers there is a beautifully simple test: start with 4, repeatedly square and subtract 2, and look at the remainder. Volunteers running this test on home computers have found every record prime of recent decades.

Proof of sufficiency

Let \(\omega=2+\sqrt3\) and \(\bar\omega=2-\sqrt3\), so \(\omega\bar\omega=1\). By induction \(S_k=\omega^{2^k}+\bar\omega^{2^k}\), since \(\big(\omega^{2^k}+\bar\omega^{2^k}\big)^2-2=\omega^{2^{k+1}}+\bar\omega^{2^{k+1}}\).

Suppose \(M_p\mid S_{p-2}\) but \(M_p\) is composite, and let \(q\) be a prime factor with \(q^2\le M_p\). Work in the finite ring \(R=\mathbb Z[\sqrt3]/(q)\), which has \(q^2\) elements, and in its group of units \(R^\times\), of order at most \(q^2-1\). In \(R\),

So \(\omega\) is a unit whose order is exactly \(2^p\) (it divides \(2^p\) but not \(2^{p-1}\)). Then \(2^p\le|R^\times|\le q^2-1 \lt M_p \lt 2^p\), a contradiction. \(\square\)

Necessity uses the arithmetic of \(\mathbb F_{M_p^2}\) and the facts that \(3\) is a quadratic non-residue and \(2\) a residue modulo \(M_p\). Each step of the test is one squaring of a number with \(p\) bits, which fast Fourier transform multiplication makes practical even for \(p\) in the hundreds of millions.

The modern search

Derrick Henry Lehmer and Raphael Robinson ran the test on the SWAC computer in 1952 and found five new Mersenne primes. Since 1996 the Great Internet Mersenne Prime Search (GIMPS) has coordinated volunteers worldwide. The largest known prime, \(M_{136\,279\,841}\), with 41,024,320 digits, was found on 12 October 2024 on cloud hardware run by Luke Durant. As of 2025, 52 Mersenne primes were known. Whether there are infinitely many is open, as is whether any odd perfect number exists.

The post-box of Europe

Before the first scientific journals of 1665, news of a discovery travelled by letter. Mersenne made himself the switchboard. His edited correspondence fills seventeen volumes, published between 1932 and 1988, and when he died letters from 78 correspondents were found in his cell. Contemporaries and historians alike have called him the post-box of Europe. His contacts reached from London and the Netherlands to Italy, Poland, Transylvania and Constantinople.

The Académie Parisiensis

From the 1620s, and on a regular weekly basis by the mid-1630s, scholars met in Paris at members’ homes and later in Mersenne’s cell to read papers, propose problems and plan experiments. Members and visitors included Gilles Personne de Roberval, Étienne Pascal and his son Blaise, Pierre Petit and Gassendi. Mersenne called it the Académie Parisienne; it is often seen as a forerunner of the Académie royale des sciences founded by Colbert in 1666.

How he worked

• Circulating. He copied letters and forwarded them, often without asking the author, so that a result proved in Toulouse could reach Paris, Leiden and Florence. Fermat’s work on tangents and maxima and minima became known largely this way.
• Provoking. He posed problems and set correspondents against each other to compare their methods. He drew attention to the cycloid, whose area Roberval found in the 1630s, and stirred the bitter dispute in 1637–38 between Descartes on one side and Fermat, Roberval and Jean de Beaugrand on the other over tangents and the Dioptrique. Historians such as Catherine Goldstein have shown that such challenges were a routine mechanism of the network.
• Publishing. He brought out other men’s work: Galileo’s mechanics, a treatise on optics by Thomas Hobbes, Samuel Sorbière’s French translation of Hobbes’s De Cive, and Roberval’s mechanics.
• Defending. He protected Descartes’s reputation among theologians and circulated the manuscript of the Meditations. He collected the objections published with it in 1641 and wrote the second set himself; Hobbes, Antoine Arnauld and Gassendi wrote others.

Descartes’s side of their exchange survives in about 146 letters, Mersenne’s in only a handful. Much of what we know of Descartes’s physics, his views on Galileo, and his reasons for suppressing Le Monde after Galileo’s condemnation in 1633 comes from letters to Mersenne.

Last journeys, death, legacy, 1644–1648

In the autumn of 1644 Mersenne travelled through Provence to Italy. He met Evangelista Torricelli and saw the mercury tube that left an apparent vacuum at its top. He doubted Torricelli’s explanation in terms of the weight of the air, but on his return he urged French scholars to repeat the experiment, and his reports reached Blaise Pascal, whose series of experiments culminated in the Puy de Dôme test of September 1648. In 1646 he visited Bordeaux, where he encouraged a local circle of scholars, and in 1646–47 he corresponded with the teenage Christiaan Huygens about falling bodies and the hanging chain. He returned to Paris in 1647 and published his last book, Novarum observationum physico-mathematicarum.

In July 1648 he went to see Descartes, briefly in Paris. Soon afterwards he fell ill with an abscess on the lung. The surgeons operated but could not find the abscess; Mersenne himself remarked that they had cut too low. He died in the convent near the Place Royale on 1 September 1648, a week before his sixtieth birthday, with Gassendi beside him. He had asked that his body be opened after death for the benefit of medical knowledge. His papers, including his optics, were published by friends in 1651.

Legacy

• Acoustics: the laws of the vibrating string, the first absolute frequency estimate, an early measurement of the speed of sound and the first reports of string harmonics. He is often called the father of acoustics.
• Music: Harmonie universelle remains a major source on seventeenth-century instruments, performance and tuning, including one of the first full treatments of equal temperament.
• Mechanics: the dissemination of Galileo in France, falling-body and pendulum measurements, and the observation that pendulum swings are not isochronous.
• Number theory: the primes \(2^p-1\), the Lucas–Lehmer test they inspired, and the GIMPS project.
• Institutions: the Académie Parisienne and his correspondence prepared the way for the Académie royale des sciences and for scientific journals.
• Optics: in Harmonie universelle he sketched designs for reflecting telescopes with two curved mirrors, though he never built one.

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Marin Mersenne, MacTutor History of Mathematics.
2. P. Hamou, Marin Mersenne, Stanford Encyclopedia of Philosophy.
3. Marin Mersenne, Wikipedia.
4. H. de Coste, La Vie du R. P. Marin Mersenne, Paris, 1649.
5. R. Lenoble, Mersenne ou la naissance du mécanisme, Vrin, 1943.
6. A. Beaulieu, Mersenne, le grand Minime, Fondation Dosne-Thiers, 1995.
7. P. Dear, Mersenne and the Learning of the Schools, Cornell University Press, 1988.
8. Correspondance du P. Marin Mersenne, religieux minime, ed. C. de Waard and others, 17 vols., Paris, 1932–1988.

Science

1. M. Mersenne, Harmonie universelle, Paris, 1636–37. English translation of the books on instruments by R. E. Chapman, Nijhoff, 1957.
2. Mersenne’s laws, Wikipedia.
3. Speed of sound, Wikipedia (history section).
4. C. Goldstein, Routine controversies: mathematical challenges in Mersenne’s correspondence, arXiv:1209.5376.
5. How did Fermat discover his theorem?, arXiv:2502.11165 (on the 1640 letters and \(M_{37}\)).
6. Mersenne prime, Wikipedia.
7. Great Internet Mersenne Prime Search (GIMPS).
8. The Prime Pages: Mersenne primes, history, theorems and lists.
9. S. Drake, The rule behind “Mersenne’s numbers”, Physis 13 (1971).