Perfect, Prime, and Sierpi ń ski Numbers

1. Perfect, Prime, and Sierpiński Numbers A mathematical excursion from the time of Pythagoras to the computer age Lane Community College Academic Colloquium – April 9, 2009 Phil Moore

2. Euclid’s Elements, ~300 B.C. Book VII, Definition 22: “A perfect number is that which is equal to its own parts.” Examples: 6 = 3 + 2 + 1 28 = 14 + 7 + 4 + 2 + 1

3. Source of the concept • Plato’s Theaetetus contains a section indicating that the idea predates Euclid. • Later tradition credits the Pythagoreans, but Aristotle documents a different use by them of the term perfect number. • The unit fractions of the Egyptians have also been suggested as a source: 1/2 + 1/3 + 1/6 = 1, for example.

4. Euclid’s Elements, Book IX, Proposition 36 “If as many numbers as we please, beginning from a unit be set out continuously in double proportion, until the sum of all becomes prime, and if the sum multiplied into the last make some number, the product will be perfect.” Illustration in proof: 1 + 2 + 4 + 8 + 16 = 31 is prime, so 31 x 16 = 496 is perfect.

5. Euclid’s formula: If 2n–1 is prime, then 2n-1(2n–1) is perfect.

6. Eratosthenes, ~250 B.C. • Showed how to systematically produce tables of primes using a “sieve”. • Presumably could have easily discovered that 27-1 = 127 was prime, thus proving that 26(27-1) = 8128 was the fourth perfect number.

7. The Neo-Pythagoreans Philo Judaeus, The Creation of the World, (c. 30 A.D.): “It was fitting, therefore, that the world, being the most perfect of created things, should be made according to the perfect number, namely, six.”

8. Nicomachus, Introduction to Arithmetic, (c. 100 A.D.) “It comes about that even as fair and excellent things are few and easily enumerated, while ugly and evil ones are widespread, so also the superabundant and deficient numbers are found in great multitude and irregularly placed – for the method of their discovery is irregular – but the perfect numbers are easily enumerated and arranged with suitable order; for only one is found among the units, 6, only one other among the tens, 28, and a third in the rank of the hundreds, 496 alone, and a fourth within the limits of the thousands, that is, below ten thousand, 8128. And it is their accompanying characteristic to end alternately in 6 or 8, and always to be even.”

9. Table of factors of 2n–1, for n to 10 • 21 –1 = 1 • 22 –1 = 3 prime • 23 –1 = 7 prime • 24 –1 = 15 = 3 · 5 • 25 –1 = 31 prime • 26 –1 = 63 = 3 · 3 · 7 • 27 –1 = 127 prime • 28 –1 = 3 · 5 · 17 • 29 –1 = 511 = 7 · 73 • 210 –1 = 1023 = 3 · 11 · 31 Note: n is prime when 2n–1 is prime!

10. Arabic mathematicians • Ibn al-Haytham (Alhazen, 965-1039) attempted to show that all even perfect numbers were of Euclid’s form. • Ibn Fallus (1194-1252) claimed that Euclid’s formula gave primes for n = 2, 3, 5, 7, 9, 11, 13, 17, 19, and 23.

11. Italians and Germans Regiomontanus and anonymous codices (c. 1458-1461): n = 2, 3, 5, 7, 13, 17 give the first six perfect numbers according to Euclid’s formula. Case of 13 was justified. 217–1 = 131,071 would have required 72 divisions to prove it prime. It was also noted that 211–1 = 2047 was equal to 23·89 and was therefore not prime.

12. Cataldi (1548-1626) Proved that n must be prime and used a table of all primes up to 750 to prove that n = p = 2, 3, 5, 7, 13, 17, and 19 generate the first seven perfect numbers. 219–1 = 524,287 required 128 divisions by all the primes up to 719 to prove it is prime.

13. Pierre de Fermat (1601-1665) • Discovered that all possible factors of 2p–1 for p prime must be of the form 2kp + 1 and found factors for p = 23, 37, and possibly 29, eliminating these as possible perfect number generators. • What about p = 31?

14. Marin Mersenne (1588-1648) • Claimed that 2p– 1 was prime for p = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, 257, and for no other numbers in this range. • His conjecture resulted in a prime of the form 2p– 1 being named a Mersenne prime.

15. Leonhard Euler (1707-1783) • Showed that factors of 2p– 1 must leave a remainder of 1 or 7 upon division by 8, which reduced the number of possible factors by roughly half. • He then proved that 231 – 1 is prime by testing all 84 possible prime factors.

16. Euler also proved that all even perfect numbers were given by Euclid’s formula. Descartes had said he saw no reason that an odd perfect number could not exist, but Euler discovered some strong constraints on the form of any such number.

17. Édouard Lucas (1842-1891) • Invented a primality testing method for Mersenne numbers in 1876 that did not require testing all possible factors. • Computed that p = 127 resulted in a Mersenne prime. • Computed that p = 67 resulted in a composite, but the composite character of 267 – 1 was not considered settled until 1894.

18. Between 1883 and 1914, the Mersenne primes for p = 61, 89, and 107 were discovered, resulting in a total of 12 known Mersenne primes and 12 known perfect numbers.

19. Derrick H. Lehmer (1905-1991) • Refined Lucas’ test, now known as the Lucas-Lehmer primality test for Mersenne numbers. • Lehmer and his wife, Emma Trotskaia Lehmer, proved in 1932 that 2257 – 1, the last number on Mersenne’s list, was actually composite.

20. The Lucas-Lehmer test • S1 = 4 • S2 = 42 – 2 = 14 • S3 = 142 – 2 = 194 • S4 = 1942 – 2 = 37634, etc. • For p ≥ 3 a prime, 2p – 1 is prime if and only if Sp-1 is divisible by 2p – 1. • Example: 25 – 1 = 31 so since S4 = 37634 is divisible by 31, 31 must be prime.

21. Considerations for efficient Lucas-Lehmer testing • The Sn grow extremely rapidly, with roughly double the number of digits at each iteration. • This can be dealt with by dealing only with the remainders: 194 ÷ 31 = 6, remainder 8, so 82 – 2 = 62, divisible by 31. • Multiplying or squaring numbers with millions of digits can be done efficiently using Fast Fourier Transforms (FFT).

22. Dawn of computer age • All p up to 257 were settled. • Max Newman and Alan Turing tested all p up to 509 on the University of Manchester Mark I computer in 1951 without finding any more Mersenne primes.

23. Raphael Robinson (1912-1995) • Used the SWAC computer at UCLA between January and October of 1952. • Discovered 5 new Mersenne primes for p = 521, 607, 1279, 2203, and 2281. • Brought the total number of known Mersenne primes to 17.

24. By 1996, there were 34 known Mersenne primes, with the last eight discoveries made on supercomputers.

25. Great Internet Mersenne Prime Search (GIMPS) • Launched in 1996 by George Woltman. • Over 100,000 participants. • Assignments coordinated by the PrimeNet server. • Has discovered 12 new Mersenne primes in 13 years.

26. Largest known prime: 243,112,609 – 1 • Discovered August 23, 2008. • Contains 12,978,189 decimal digits. • Verified using multi-processor machines. • The associated perfect number, 243,112,608(243,112,609 – 1), contains 25,956,377 digits! • Claimed the EFF $100,000 prize for the first proven prime of over ten million digits.

29. Odd perfect numbers? • The question of their existence has been called the oldest unsolved math problem. • Must contain over 300 digits. • Must contain at least 75 prime factors. • Must contain at least 9 distinct prime factors. • Heuristic arguments suggest that none exist, but the question is still open.

30. Fermat primes • Fermat knew that for 2n+1 to be prime, n must be a power of 2: • 21 + 1 = 3 prime • 22 + 1 = 5 prime • 24 + 1 = 17 prime • 28 + 1 = 257 prime • 216 + 1 = 65537 prime • Fermat thought that these numbers 22m + 1 were always prime!

31. WRONG! • Euler proved in 1732 that 232 +1, the “fifth” Fermat number, was composite: • 232 +1 = 4,294,967,297 = 641 · 6,700,417. • We now know that the 5th through the 32nd Fermat numbers are all composite, as well as over 200 larger Fermat numbers. • Most of these numbers have been proven composite through finding factors.

32. Euler and Lagrange: Any factor of a Fermat number 22m + 1 must be of the form k·2n + 1 where n ≥ m + 2. • Early researchers noted that some k values gave sequences of k·2n + 1 that were rich in primes, other k values gave sequences very sparse in primes.

33. Waclaw Sierpiński (1882-1969) Proved in 1960 that there are infinitely many positive odd integer values of k such that k·2n + 1 is composite for any positive integer n.

34. John Selfridge proved in 1962 that k = 78557 is an example of such a Sierpiński number, and raised the question of whether it was the smallest. • It can be easily proven that for all n, 78557·2n+1 is always divisible by at least one number in the finite “covering set” {3,5,7,13,19,37,73}.

35. Paul Erdős (1913-1996) • Conjectured that any Sierpiński number must have a finite covering set. • Recent evidence indicates that his conjecture is probably false for certain values of k which are perfect powers. • It is still believed that 78557 is the smallest Sierpiński number.

36. The Sierpiński Problem • For each positive odd integer k < 78557, find a positive integer n such that k·2n+1 is prime. • The distributed computing project Seventeen or Bust was started in 2002 to work on the remaining 17 k values. • To date, six k values are still unresolved.

37. The dual Sierpiński problem • Replace n by a negative integer: k·2-n+1 = (k + 2n) / 2n. • Again, 2n +78557 is always composite with the same covering set {3,5,7,13,19,37,73}. • Is k = 78557 the smallest positive odd integer with this property?

38. Dual Sierpiński investigation: • For each positive odd k < 78557, find an n such that k + 2n is prime. • Of these 39,278 values of k, a prime value of k + 2n is known for all but 33 of them. • For 30 of these 33 remaining k values, a probable prime value of k + 2n is known. • The three remaining sequences are being searched by “Five or Bust”.

39. The largest known prime numbers number digits year notes • 2^43112609-1 12978189 2008 Mersenne 46? • 2^37156667-1 11185272 2008 Mersenne 45? 3 2^32582657-1 9808358 2006 Mersenne 44? 4 2^30402457-1 9152052 2005 Mersenne 43? 5 2^25964951-1 7816230 2005 Mersenne 42? • 2^24036583-1 7235733 2004 Mersenne 41? 7 2^20996011-1 6320430 2003 Mersenne 40? 8 2^13466917-1 4053946 2001 Mersenne 39 9 19249*2^13018586+1 3918990 2007 (Seventeen or Bust) 10 27653*2^9167433+1 2759677 2005 (Seventeen or Bust) 11 28433*2^7830457+1 2357207 2004 (Seventeen or Bust) 12 3661*2^7031232+1 2116617 2007 (Seventeen or Bust) 13 2^6972593-1 2098960 1999 Mersenne 38 • 258317*2^5450519+1 1640776 2008 • 3*2^5082306+1 1529928 2009 16 5359*2^5054502+1 1521561 2003 (Seventeen or Bust) 17 265711*2^4858008+1 1462412 2008 18 3*2^4235414-1 1274988 2008 19 24518^262144+1 1150678 2008 20 938237*2^3752950-1 1129757 2007 Note: All numbers on this list are of the form N±1.

40. Proven primes • Proofs are easier if we know all or at least many of the factors of N+1 or N-1. • For large numbers which are not of such a “special form”, methods of proof are much more difficult. • The largest number of general form which has so far been proven prime has 20,562 digits.

41. Resolution of the Mixed Sierpiński Problem • Paper published by INTEGERS online journal • Authors: Louis Helm, Phil Moore, Payam Samidoost, and George Woltman • Abstract: Recent progress on the Sierpiński problem has resulted in the following theorem: 78557 is the smallest positive odd integer k such that both k·2n+1 and k+2n are composite for any positive integer n. An algorithmic enhancement to the fast Fourier transform routines used in this research is described. Prospects for the eventual resolution of both the original and the dual Sierpiński problems are estimated.

42. Probable primes • Pass tests that all prime numbers will pass and most composite numbers will fail. • Term is usually used for numbers which are not proven primes. • Called “industrial grade” primes in cryptology.

43. Large probable primes discovered in this dual Sierpiński investigation: • 21191375 + 8543, discovered June 2008 at LCC, at 358,640 digits was the record holder until October. • 21518191 + 75353, discovered January 4, 2009 by Five or Bust, at 457,022 digits held the record for a short time. • 22249255 + 28433, discovered January 26, 2009 by Five or Bust, at 677,094 digits is the current record holder.

44. Fact sheet on 22249255 + 28433 • The probability that this record probable prime is actually composite is less than one in 10900. • To prove that it is actually prime would take an estimated 3 billion years. • If the Generalized Riemann Hypothesis is ever proven, we could prove it is prime in just one year using 3 billion computers!

45. Five or Bust • Begun in October 2008 to search the remaining 5 sequences 2n + k. • Sieving removes candidates divisible by a “small” factor (now up to 150 trillion or so.) • Each remaining candidate is subjected to a probable prime test. • The unsolved sequences correspond to the values k = 2131, 40291, and 41693.

47. Current search limits on n for given probabilities of solution

48. What about k·2n – 1 ? • Hans Riesel (1956): There are infinitely many values of k such that k·2n – 1 is always composite. • One such value is k = 509203, as the sequence 509203·2n – 1 has the covering set {3,5,7,13,17,241}. Is 509203 the smallest such value of k? • Currently 64 odd values of k < 509203 are unsettled.

49. What about 2n – k ? • Replace n by -n again, and see that k·2-n – 1 = (k – 2n) / 2n. • k – 2n can be positive or negative, so take the absolute value. • If k = 509203, |2n – 509203| has a covering set and is therefore always composite. Is k = 509203 the smallest such value of k?

50. Current status of 2n – k • All n searched up to 262,000. • 87 values of k < 509203 are still unresolved. Another distributed search? • Note: 509203 is about six and a half times larger than 78557, so the Riesel problem and the dual Riesel problem are quite a bit larger than the Sierpiński problem and its dual. These problems may never be resolved within our lifetimes!