Download Perfect Numbers: An Elementary Introduction and more Lecture notes Number Theory in PDF only on Docsity!
PERFECT NUMBERS: AN ELEMENTARY INTRODUCTION
JOHN VOIGHT
Abstract. This serves as an elementary introduction to the history and the- ory surrounding even perfect numbers.
One would be hard put to find a set of whole numbers with a more fascinating history and more elegant properties surrounded by greater depths of mystery—and more totally useless—than the perfect numbers. —Martin Gardner [2] The number 6 is unique in that 6 = 1+2+3, where 1, 2, and 3 are all of the proper divisors of 6. The number 28 also shares this property, for 28 = 1 + 2 + 4 + 7 + 14. These “perfect” numbers have seen a great deal of mathematical study—indeed, many of the basic theorems of number theory stem from the investigation of the Greeks into the problem of perfect and Pythagorean numbers [16]. Moreover, it was while investigating these numbers that Fermat discovered the (little) theorem that bears his name and which forms the basis of a substantial part of the theory of numbers. Though it is rooted in ancient times, remarkably this subject remains very much alive today, harboring perhaps the “oldest unfinished project of mathematics” [17]. This paper surveys the history and elementary results concerning perfect num- bers.
- Early History The almost mystical regard for perfect numbers is as old as the mathematics concerning them. The Pythagoreans equated the perfect number 6 to marriage, health, and beauty on account of the integrity and agreement of its parts. Around 100 c.e., Nicomachus noted that perfect numbers strike a harmony between the extremes of excess and deficiency (as when the sum of a number’s divisors is too large or small), and fall in the “suitable” order: 6, 28, 496, and 8128 are the only perfect numbers in the intervals between 1, 10, 100, 1000, 10000, and they end alternately in 6 and 8. Near the end of the twelfth century, Rabbi Josef b. Jehuda Ankin suggested that the careful study of perfect numbers was an essential part of healing the soul. Erycius Puteanus in 1640 quotes work assigning the perfect number 6 to Venus, formed from the triad (male, odd) and the dyad (female, even). Hrotsvit, a Benedictine in the Abbey of Gandersheim of Saxony and perhaps the earliest female German poet, listed the first four perfect numbers in her play Sapientia as early as the tenth century:
Date: May 31, 1998. 1
2 JOHN VOIGHT
We should not leave unmentioned the principal numbers... those which are called “perfect numbers.” These have parts which are nei- ther larger nor smaller than the number itself, such as the number six, whose parts, three, two, and one, add up to exactly the same sum as the number itself. For the same reason twenty-eight, four hundred ninety-six, and eight thousand one hundred twenty-eight are called perfect numbers [2]. Saint Augustine (among others, including the early Hebrews) considered 6 to be a truly perfect number—God fashioned the Earth in precisely this many days (rather than at once) to signify the perfection of His work. Indeed, as recorded by Alcuin of York (who lived from 732 to 804 c.e.), the second origin was imperfect, as it arose from the deficient number 8 > 1 + 2 + 4, this number counting the 8 souls in Noah’s ark (Noah, his three sons, and their four wives, in Genesis, chapter 7) from which sprung the entire human race [8]. Philo Judeus, in the first century c.e., called 6 the most productive of all numbers, being the smallest perfect number. Throughout the centuries that followed, various mathematicians carefully stud- ied perfect numbers (see the continued extensive history given by Dickson [6] and also by Picutti [14]). Up to the time of Descartes and Fermat, a sizeable pool of important results—as well as much misinformation—had been collected.
- Elementary Results We now turn to some definitions and some elementary results.
Definition 1. The sum of divisors is the function σ(n) =
d|n d, where^ d^ runs over the positive divisors of n including 1 and n itself.
For example, σ(11) = 1 + 11 = 12 and σ(15) = 1 + 3 + 5 + 15 = 24.
Definition 2. The number N is said to be perfect if σ(N ) = 2N. When σ(N ) < 2 N , we say N is deficient; when σ(N ) > 2 N , we say N is abundant.
The definition of perfect is equivalent to saying that the sum of the proper (or aliquot) divisors of N is equal to N (we just do not add N itself to the sum). While this may seem more natural, the central reason for using the function σ is that it possesses some very special properties. First:
Proposition 3. σ(mn) = σ(m)σ(n) whenever gcd(m, n) = 1.
Proof. If d is a divisor of mn, then by unique factorization we can represent d uniquely as the product of a divisor of m and a divisor of n (since they have no common factors). That is, every term of σ(mn) (the sum of all divisors of mn) occurs exactly once in the sum σ(m)σ(n) (the product of all divisors of m and n). The converse is also true: every such product is a divisor of mn, so the sums must be the same. This suffices to establish the proposition. It might be easier also just to appeal to the very special way in which σ is defined as a divisor sum (for the details consult [12]).
Hence σ is completely determined when its value is known for every prime-power argument. (Such functions are, of course, the multiplicative functions.) This yields the very useful:
4 JOHN VOIGHT
- Even Perfect Numbers The first individual to really categorize the perfect numbers was the Greek math- ematician Euclid. He noticed that the first four perfect numbers are of a very specific form:
6 = 2^1 (1 + 2) = 2 · 3 ,
28 = 2^2 (1 + 2 + 2^2 ) = 4 · 7 ,
496 = 2^4 (1 + 2 + 2^2 + 2^3 + 2^4 ) = 16 · 31 , and 8128 = 2^6 (1 + 2 + · · · + 2^6 ) = 64 · 127.
Notice, though, that the numbers 90 = 2^3 (1 + 2 + 2^2 + 2^3 ) = 8 · 15 and 2016 = 25 (1 + 2 + · · · + 2^5 ) = 32 · 63 are missing from this sequence. As Euclid pointed out, this is because 15 = 3 · 5 and 63 = 3^2 · 7 are both composite, whereas the numbers 3, 7, 31, 127 are all prime. As it appears in Book IX, proposition 36 of his Elements, Euclid writes: “If as many numbers as we please beginning from an 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.” We state this observation in a slightly more compact form:
Theorem 7 (Euclid). If 2 n^ − 1 is prime, then N = 2n−^1 (2n^ − 1) is perfect.
Proof. Clearly the only prime divisors of N are 2n^ − 1 and 2. Since 2n^ − 1 occurs as a single prime, we have simply that σ(2n^ − 1) =
1 + (2n^ − 1)
= 2n, and thus
σ(N ) = σ(2n−^1 )σ(2n^ − 1) =
2 n^ − 1 2 − 1
2 n^ = 2n(2n^ − 1) = 2N.
So N is perfect.
The task of finding perfect numbers, then, is intimately linked with finding primes of the form 2n^ − 1. Such numbers are referred to as Mersenne primes, after the seventeenth century monk Marin Mersenne, a colleague of Descartes, Fermat, and Pascal. He is credited with investigating these unique primes as early as 1644. Mersenne knew that 2n^ − 1 is prime for n = 2, 3, 5, 11, 13, 17, and 19—and, more brilliantly, conjectured the cases n = 31, 67, 127, 257. It took nearly two hundred years to test these numbers. There is one important criterion that hones in on the primality of Mersenne numbers:
Proposition 8 (Cataldi-Fermat). If 2 n^ − 1 is prime, then n itself is prime.
Proof. Note from before, xn^ − 1 = (x − 1)(xn−^1 + · · · + x + 1). Suppose we can write n = rs, where r, s > 1. Then
2 n^ − 1 = (2r^ )s^ − 1 = (2r^ − 1)
(2r^ )s−^1 + · · · + 2r^ + 1
so that (2r^ − 1) | (2n^ − 1) which is prime, a contradiction.
Note that the converse is not true—the number 2^11 − 1 = 2047 = 23 · 89, for instance.
PERFECT NUMBERS 5
Must all perfect numbers be of Euclid’s type? Leonard Euler, in a posthumous paper, proved that every even perfect number is of this type. Many ingenious proofs of this fact exist.
Theorem 9 (Euler). If N is an even perfect number, then N can be written in the form N = 2n−^1 (2n^ − 1), where 2 n^ − 1 is prime.
Proof. The first proof is Euler’s own [6]. Let N = 2n−^1 m be perfect, where m is odd; since 2 does not divide m, it is relatively prime to 2n−^1 , and
σ(N ) = σ(2n−^1 m) = σ(2n−^1 )σ(m) =
2 n^ − 1 2 − 1
σ(m) = (2n^ − 1)σ(m).
N is perfect so σ(N ) = 2N = 2(2n−^1 m) = 2nm, and with the above, 2nm = (2n^ − 1)σ(m). Let s = σ(m). We then have m = (2n^ − 1)(s/ 2 n); since 2n^ does not divide 2 n^ − 1, it must divide s (because m is an integer), so that m = (2n^ − 1)q for some q = s/ 2 n. If q = 1, we have a number of Euclid’s type, for then m = 2n^ − 1 and s = σ(m) = 2n^ = (2n^ − 1) + 1 = m + 1. Since σ(m) is the sum of all of the divisors of m, m = 2n^ − 1 must be a prime, and N = 2n−^1 m = 2n−^1 (2n^ − 1). If q > 1, we retotal the sum of the divisors of m = (2n^ − 1)q. The factors of m then include 1, q, 2n^ − 1, and m itself, so that
s = σ(m) ≥ 1 + q + (2n^ − 1) + (2n^ − 1)q =
(2n^ − 1) + 1)(q + 1) = 2n(q + 1).
But this implies
m s
(2n^ − 1)q 2 n(q + 1)
2 n^ − 1 2 n
q q + 1
2 n^ − 1 2 n^
an impossibility, for we have previously established equality: σ(N ) = 2nm = (2n^ − 1)s implies that m/s = (2n^ − 1)/ 2 n.
Proof 2. A second, simpler proof is given by Dickson [5]. From 2nm = (2n^ −1)σ(m) we note
σ(m) = 2 nm 2 n^ − 1
(2n^ − 1) + 1
m 2 n^ − 1
= m + m 2 n^ − 1
Since both σ(m) and m are integers, so too must d = m/(2n^ − 1) be an integer. Thus (2n^ − 1) | m and consequently d itself divides m. But σ(m) = m + m/(2n^ − 1) = m + d is the sum of all of the divisors of m. How can this be? Certainly 1 divides m, so are forced to conclude that d = 1; if this were not the case, then we would have σ(m) = m + d + 1, a contradiction. Therefore, m = 2n^ − 1, and in partiular, m has no other positive divisors other than one and itself, so 2n^ − 1 is prime.
Proof 3. A third proof is given by McDaniel [10]. Since 2nm = (2n^ − 1)σ(m), every prime divisor of 2n^ − 1 must also divide m (for it is odd and cannot divide 2n). So suppose pα^ divides 2n^ − 1 with p prime. By Lemma 5, σ(m) m
σ(pα) pα^
1 + p + · · · + pα pα^
pα−^1 + pα pα^
1 + p p
Hence,
1 =
σ(N ) 2 N
σ(2n−^1 )σ(m) 2 nm
(2n^ − 1)(1 + p) 2 np
(2n^ − 1) − p 2 np
PERFECT NUMBERS 7
and therefore
2 α^1 +
∏^ k
i=
pα i i−γi=
∏^ k
i=
pγ ii
) (^) k ∏
i=
pα i i−γi=
∏^ k
i=
(1 + pi + · · · + pα i i)
so
pα 2 2 −γ^2 pα 3 3 −γ^3 · · · pα k k^ −γk+pα 2 2 pα 3 3 · · · pα k k= (1+p 2 +· · ·+pα 2 2 ) · · · (1+pk +· · ·+pα k k).
Since 0 ≤ γi ≤ αi for all i, expanding the product on the right will yield each of the terms on the left. But there are the only two, which means that γi = 0 for all values of i except one; so then k = 2, and α 2 = 1. The above equation then yields simply p^12 − γ^2 + p 2 = 1 + p 2 or p^12 − γ^2 = 1,
so γ 2 = 1, and N is of Euclid’s type as claimed.
Even perfect numbers have a number of nice little properties. We say that a number is triangular if it can be arranged in triangular fashion, that is, one atop two atop three (and so on) arranged in a triangular lattice. While perfect numbers can be written in a geometric expression `a la Euclid, they may also be derived from arithemetic progressions:
6 = 1 + 2 + 3, 28 = 1 + 2 + 3 + 4 + 5 + 6 + 7, 496 = 1 + 2 + 3 + · · · + 31.
We state this result as follows:
Proposition 10. If N is an even perfect number, then N is triangular.
Proof. We have m triangular if m =
∑k− 1 i=1 i^ = 1 + 2 +^ · · ·^ +^ k^ = (^
1 2 )k(k^ −^ 1) for some k. But note that N = 2n−^1 (2n^ − 1) = ( 12 )2n(2n^ − 1).
We can also write any perfect number as the sum of cubes:
Proposition 11 ([1]). If N = 2n−^1 (2n^ − 1) is perfect then n = 1^3 + 3^3 + · · · + ( 2 (n−1)/^2 − 1
Proof. Recall the formula ∑n
i=
i^3 =
n^2 (n + 1)^2 4
a fact which can be proved by induction. Let m = 2(n−1)/^2 ; we have then that
13 + 3^3 + · · · + (2m − 1)^3 =
13 + 2^3 + · · · + (2m)^3
23 + 4^3 + · · · + (2m)^3
(2m)^2 (2m + 1)^2 4
m^2 (m + 1)^2 4 = m^2 (2m + 1)^2 − 2 m^2 (m + 1)^2 = m^2 (4m^2 + 4m + 1 − 2 m^2 − 4 m − 1) = m^2 (2m^2 − 1).
By substituting for m we get the desired result.
Perfect numbers have very unique representations. As first noted by Studniˇcka, we have 6 = 110 2 , 28 = 11100 2 , 496 = 111110000 2 , and 8128 = 1111111000000 2 written in binary; in general:
Proposition 12. If N = 2n−^1 (2n^ − 1) is perfect and N is written in base 2 , then it has 2 n − 1 digits, the first n of which are unity and the last n − 1 are zero.
8 JOHN VOIGHT
Proof. This follows immediately from how numbers are written in binary, by re- calling that 2p^ − 1 = 1 + 2 + · · · + 2p−^1.
We also have the historically interesting fact:
Proposition 13. Every even perfect number ends in either 6 or 8.
Proof. Every prime greater than 2 is of the form 4m + 1 or 4m + 3. In the first case,
N = 2n−^1 (2n^ − 1) = 2^4 m(2^4 m+1^ − 1) = 16m(2 · 16 m^ − 1) ≡ 6 m(2 · 6 m^ − 1) ≡ 6(12 − 1) ≡ 6 (mod 10),
since by induction it is clear that 6m^ ≡ 6 (mod 10) for all m. Similarly, in the second case,
N = 4 · 16 m(8 · 16 m^ − 1) ≡ 4 · 6(8 · 6 − 1) ≡ 4(8 − 1) ≡ 8 (mod 10).
Finally, if n = 2, N = 6, and so we have captured all the possibilities.
Even perfect numbers exhibit another amazing property, first proven in 1844:
Proposition 14 (Wantzel). The iterative sum of the digits of an even perfect number (other than 6) is one.
Proof. Let N = 2n−^1 (2n^ − 1). The first claim we establish is that N ≡ 1 (mod 9). Now n 6 ≡ 0 (mod 3), for then 3 | n so n is not prime (N 6 = 6) and N is not perfect. Hence N ≡ 1 or 2 modulo 3. Suppose we have the first case, and n = 3k + 1 for some k. Now n is odd, so k is even and we could have written n = 6k + 1. Hence
N = 2^6 k(2^6 k+1^ − 1) = 64k(2 · 64 k^ − 1) ≡ (1)
= 1 (mod 9). In the second case, n = 3k + 2 for some k, so as n is not even, k is odd and n = 6k + 5. Thus,
N = 2^6 k+4(2 · 26 k+5^ − 1) ≡ 16(1)
≡ (−2)(− 4 − 1) = 10 ≡ 1 (mod 9). The work is now done, for the iterative sum of the digits of a number is merely its congruence class mod 9. For if N = ar... a 1 a 0 , where the ai are digits, then the sum of digits is ∑r
i=
ai 10 i^ ≡
∑^ r
i=
ai 1 i^ =
∑^ r
i=
ai (mod 9).
By iterating this sum we get the desired result.
Historically, there are a number of problems that arise in the investigation of even perfect numbers. For instance, Fermat declared that if n is a prime, then 2 n^ − 2 is divisble by 2n. Of course, this immediately follows from the fact that 2 n−^1 ≡ 1 modulo n if n is prime by Fermat’s little theorem. Fermat also showed that if n is prime, then 2n^ − 1 is divisible by no prime other than those of the form 2kn + 1. If q | (2n^ − 1) for some prime q 6 = 2, then 2n^ ≡ 1 modulo q. Since we have a power of 2 equal to 1, the identity, the order of 2 divides n; and since n is prime, we have the order just equal to n. But also n | (q − 1), the order of the multiplicative group. q is odd so q − 1 is even and we have even better that 2n | (q − 1), or 2nk = q − 1, and the result follows. Furthermore, Philiatrus discovered two primes n < 50 where N = 2n^ − 1 is not prime: 2^23 − 1 is divisible by 47 and 2^41 − 1 is divisble by 83. Lucas proved the following: if p = 8m + 7 is a prime, then 2^4 m+3^ − 1 is not. It can be shown that if
10 JOHN VOIGHT
The Mersenne primes appear to be regularly distributed—“ballpark” estimates (meaning within thousands of orders of magnitude) can be given [15], but ultimately these numbers must be checked out one by one. We close with one interesting result:
Conjecture 15 (Cunningham). If p = 2x^ ± 1 or 2 x^ ± 3 is prime, p ≡ 3 (mod 4), and 2 p + 1 is a prime, then 2 p^ − 1 is prime.
All known and conjectured primes 2p^ − 1, with p prime, fall under this rule.
References
- Syed Asadulla, Even perfect numbers and their Euler’s function, Internat. J. Math. Math. Sci. 10 (1987), 409–412.
- Stanley J. Bezuszka and Margaret J. Kenney, Even perfect numbers: (update)^2 , Math. Teacher 90 (1997), 628–633.
- R. D. Carmichael, Multiply perfect numbers of four different primes, Annals of Math. 8 (1906- 1907), 149–158.
- Graeme L. Cohen, Even perfect numbers, Math. Gaz. 65 (1981), 28–30.
- L. E. Dickson, Notes on the theory of numbers, Amer. Math. Monthly 18 (1911), 109.
- Leonard Eugene Dickson, History of the theory of numbers, vol. 1, pp. 3–33, Chelsea Pub. Co., New York, 1971.
- I.N. Herstein, Abstract algebra, 3rd ed., Prentice-Hall, Upper Saddle River, N.J., 1996.
- David G. Kendall, The scale of perfection, J. Appl. Prob. 19A (1982), 125–138.
- J. Knopfmacher, A note on perfect numbers, Math. Gazette 44 (1960), 45.
- Wayne L. McDaniel, On the proof that all even perfect numbers are of Euclid’s type, Math. Mag. 48 (1975), 107–108.
- Mersenne prime search, http://www.mersenne.org/prime.htm, 1998.
- Ivan Morton Niven, Herbert S. Zuckerman, and Hugh L. Montgomery, An introduction to the theory of numbers, 5th ed., John Wiley & Sons, Inc., 1991.
- Oystein Ore, Number theory and its history, pp. 91–100, McGraw-Hill, 1948.
- Ettore Picutti, Pour l’histoire des sept premiers nombres parfaits, Historia Math. 16 (1989), 123–136.
- Manfred R. Schroeder, Where is the next Mersenne prime hiding?, Math. Intelligencer 5 (1983), 31–33.
- Daniel Shanks, Solved and unsolved problems in number theory, 2nd ed., pp. 1–8, 12–15, 18–29, Chelsea Pub. Co., New York, 1978.
- Stan Wagon, Perfect numbers, Math. Intelligencer 7 (1985), 66–68. E-mail address: [email protected] Department of Mathematics, University of California, Berkeley
