History

Five interesting things about Mersenne primes

Posted on by John

A Mersenne prime is a prime number that is one less than a power of 2. These primes are indexed by the corresponding power of two, i.e. Mp = 2p − 1. It turns out p must be prime before 2p − 1 can be prime.

Here are five things I find interesting about Mersenne primes.

1. Record size primes

The largest known prime number is a Mersenne prime, M82,589,993, proved prime in 2018. And ever since M521 was proven prime in 1952, the largest known prime has always been a Mersenne prime (with one exception in 1989). See a history of prime number records.

One reason for the prevalence of Mersenne primes in the record books is that there is a special algorithm for testing whether a number of the form 2p − 1 is prime, the Lucas-Lehmer test.

2. Finiteness

There may only be a finite number of Mersenne primes. Only 51 are known so far. There are conjectures that predict there are an infinite number of Mersenne primes, but these have not been settled.

3. Connection with perfect numbers

Euclid proved that if M is a Mersenne prime, M(M + 1)/2 is a perfect number. Two millennia later, Euler proved that if N is an even perfect number, N must be of the form M(M + 1)/2 where M is a Mersenne prime. (More details here.)

Since we only know of 51 Mersenne primes at the moment, and we don’t know of any odd perfect numbers, there are only 51 known perfect numbers.

4. Connection with random number generation

The Mersenne twister is a popular, high-quality random number generator. The generator is so named because its period is a Mersenne prime, M19,937.

5. History

Mersenne primes are named after the French monk Marin Mersenne (1588–1648) who compiled a list of Mersenne primes. Mersenne wasn’t the first to be aware of such primes. As mentioned above, Euclid connected these primes with perfect numbers.

Marin Mersenne is one of my academic ancestors. I studied under Ralph Showalter, who studied under Tsuan Ting, and so forth back to Frans van Schooten Jr., who studied under Marin Mersenne.

What I find fascinating about this is not my particular genealogy, but that adequate records exist to construct such genealogies. The Mathematics Genealogy Project has over 150,000 records, some reaching back to the Middle Ages.

 

What was the most important event of the 19th century?

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According to Richard Feynman, the most important event of the 19th century was the discovery of the laws of electricity and magnetism.

From a long view of the history of mankind—seen from, say, ten thousand years from now—there can be little doubt that the most significant event of the 19th century will be judged as Maxwell’s discovery of the laws of electrodynamics. The American Civil War will pale into provincial insignificance in comparison with this important scientific event of the same decade.

From The Feynman Lectures on Physics, Volume 2.

Related post: Grand unified theory of 19th century math

Jupiter’s magic square

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Albrecht Dürer’s engraving Melencolia I contains an interesting magic square toward the top right corner.

Here’s a close-up of the magic square:

The square has the following properties:

  • Every row, column, and diagonal sums to 34.
  • The four squares in the center sum to 34.
  • The four squares in the corners sum to 34.
  • Each quadrant sums to 34.
  • The year the engraving was made, 1514, appears in the bottom row.

I’d seen all this years ago, but this week I learned something else about this square.

Magic squares of different sizes were traditionally associated with planets in the solar system. … the 4 × 4 square in Melancolia is Jupiter’s … One suggestion for Dürer’s use of the square is that it reflected the mystical belief that Jupiter’s joyfulness could counteract the sense of melancholy that pervades the engraving.

From The Number Mysteries.

Regarding “Jupiter’s joyfulness,” here’s the etymology of jovial from Online Etymology Dictionary.

1580s, from Fr., from It. joviale, lit. “pertaining to Jupiter,” from L. Jovialis “of Jupiter,” from Jovius (used as gen. of Juppiter) “Jupiter,” Roman god of the sky. The meaning “good-humored, merry,” is from astrological belief that those born under the sign of the planet Jupiter are of such dispositions. In classical L., the compound Juppiter replaced Old L. Jovis as the god’s name. Related: Jovially.

Related posts

History is strange

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From historian Patrick Allitt of Emory University:

History is strange, it’s alien, and it won’t give us what we would like to have. If you hear a historical story and at the end you feel thoroughly satisfied by it and find that it perfectly coincides with your political inclinations, it probably means that you’re actually listening to ideology or mythology. History won’t oblige us, and much of its challenge and interest comes from its immovable differentness from us and our own world.

The first FORTRAN program

Posted on by John

The first FORTRAN compiler shipped this week in 1957. Herbert Bright gives his account of running his first FORTRAN program with the new compiler here.

(Bright gives the date as Friday, April 20, 1957, but April 20 fell on a Saturday that year. It seems more plausible that he correctly remembered the day of the week—he says it was late on a Friday afternoon—than that he remembered the day of the month, so it was probably Friday, April 19, 1957.)

For more history, see Twenty Five Years of FORTRAN by J. A. N. Lee written in 1982.

Thanks to On This Day in Math for the story.

Augustine, Leibowitz, and evolution

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The following paragraph is from the science fiction novel A Canticle for Leibowitz:

A fourth century bishop and philosopher. He [Saint Augustine] suggested that in the beginning God created all things in their germinal causes, including the physiology of man, and that the germinal causes inseminate, as it were, the formless matter—which then gradually evolved into the more complex shapes, and eventually Man. Has this hypothesis been considered?

A Canticle for Leibowitz is set centuries after a nuclear holocaust. The war was immediately followed by the “Simplification.” Survivors rejected all advanced technology and hunted down everyone who was even literate. At this point in the book, a sort of Renaissance is taking place. The question above is addressed to a scientist who is explaining some of the (re)discoveries taking place. The scientist’s response was

“I’m afraid it has not, but I shall look it up,” he said, in a tone that indicated he would not.

Was the reference to Augustine simply made up for the novel, or is there something in Augustine’s writings that the author is alluding to? If so, does anyone know what in particular he may be referring to? Is such a proto-Darwinian reading of Augustine fair?

Thomas Jefferson and preparing for meetings

Posted on by John

Here’s an interesting historical anecdote from Karl Fogel’s Producing Open Source Software on the value of preparing for meetings.

In his multi-volume biography of Thomas Jefferson, Jefferson and His Time, Dumas Malone tells the story of how Jefferson handled the first meeting held to decide the organization of the future University of Virginia. The University had been Jefferson’s idea in the first place, but (as is the case everywhere, not just in open source projects) many other parties had climbed on board quickly, each with their own interests and agendas.

When they gathered at that first meeting to hash things out, Jefferson made sure to show up with meticulously prepared architectural drawings, detailed budgets for construction and operation, a proposed curriculum, and the names of specific faculty he wanted to import from Europe. No one else in the room was even remotely as prepared; the group essentially had to capitulate to Jefferson’s vision, and the University was eventually founded more or less in accordance with his plans.

The facts that construction went far over budget, and that many of his ideas did not, for various reasons, work out in the end, were all things Jefferson probably knew perfectly well would happen. His purpose was strategic: to show up at the meeting with something so substantive that everyone else would have to fall into the role of simply proposing modifications to it, so that the overall shape, and therefore schedule, of the project would be roughly as he wanted.

Related posts

Military intelligence from serial numbers

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During World War II, America and her allies needed to estimate the number of Panzer V tanks Germany had produced. The solution was simple: Look at the serial numbers of the captured tanks. If you assume the tanks had been sequentially numbered — as in fact they were — you could view the serial numbers of the captured tanks as random samples from the entire range. You could then use statistics to estimate the range and hence the number of tanks produced. More details available here.

A few years later America tried to use the serial number trick to estimate the number of Soviet strategic bombers. This time the trick backfired.

In 1958, American military intelligence believed the USSR would soon have four hundred Bison and three hundred Bear bombers capable of striking the American heartland. Their evidence was the high serial number of a Bison that had flown at a May Day parade in Moscow. In fact, the Soviets knew the Americans were watching, and intentionally inflated that number. — Rocket Men, page 118.

The Panzer estimate was accurate because the Allies had hundreds of data points, enough to support the assumption that the tanks were sequentially numbered and to make a good estimate of the total number.

The Bison bomber was only one data point, but it was consistent with what intelligence services (wrongly) believed. At that time, the US had grossly over-estimated the military capabilities of the USSR. According to Rocket Men, Khrushchev turned down US offers to cooperate in space exploration because he feared that such cooperation would give the US a more accurate assessment of his country’s military.

Related post: Selection bias and bombers