Leonardo Fibonacci died ab1250 Italian mathematician. The elements of the sequence of numbers 1, 1, 2, 3, 5, 8, 13, 21, …, each of which, after the second, is the sum of the two previous numbers. These numbers were first noted by the medieval Italian mathematician Leonardo Pisano (“Fibonacci”) in his Liber abaci (1202; “Book of the Abacus”), which also popularized Hindu-Arabic numerals and the decimal number system in Europe. Leonardo of Pisa , original name Leonardo Fibonacci. Liber Abaci (1202) is an historic book on arithmetic by Leonardo of Pisa, known later by his nickname Fibonacci. Its title has two common translations, The Book of the Abacus or The Book of Calculation. Liber Abaci was not the first Western book to describe Hindu-Arabic numerals, the first being by Pope Silvester II in 999, but by addressing tradesmen and academics, it began to convince the public of the superiority of the new numerals. The first section introduces the Hindu-Arabic numeral system. The second section presents examples from commerce, such as conversions of currency and measurements, and calculations of profit and interest. The third section discusses a number of mathematical problems; for instance, it includes (ch. II.12) the Chinese remainder theorem, perfect numbers and Mersenne primes as well as formulas for arithmetic series and for square pyramidal numbers. Another example in this chapter, describing the growth of a population of rabbits, was the origin of the Fibonacci sequence for which the author is most famous today. The fourth section derives approximations, both numerical and geometrical, of irrational numbers such as square roots. The book also includes Euclidean geometric proofs, and a study of simultaneous linear equations following Diophantus, which Fibonacci most likely learned from Arab mathematician al-Karaji (Ore 1948).
The Fibonacci numbers first appeared, under the name matrameru (mountain of cadence), in the work of the Sanskrit grammarian Pingala (Chandah-shastra, the Art of Prosody, 450 or 200 BC). Prosody was important in ancient Indian ritual because of an emphasis on the purity of utterance. The Indian mathematician Virahanka (6th century AD) showed how the Fibonacci sequence arose in the analysis of metres with long and short syllables. Subsequently, the Jain philosopher Hemachandra (c.1150) composed a well known text on these. A commentary on Virahanka by Gopala in the 12th c. also revisits the problem in some detail.
Sanskrit vowel sounds can be long (L) or short (S), and Virahanka's analysis, which came to be known as matra-v?tta wishes to compute how many metres (matras) of a given overall length can be composed of these syllables. If the long syllable is twice as long as the short, the solutions are:
1 mora: S (1 pattern)
2 morae: SS; L (2)
3 morae: SSS, SL; LS (3)
4 morae: SSSS, SSL, SLS; LSS, LL (5)
5 morae: SSSSS, SSSL, SSLS, SLSS, SLL; LSSS, LSL, LLS (8)
A pattern of length n can be formed by adding S to a pattern of length n-1, or L to a pattern of length n-2; and the prosodicists showed that the number of patterns of length n is the sum of the two previous numbers in the series. Donald Knuth reviews this work in The Art of Computer Programming as equivalent formulations of the bin packing problem for items of lengths 1 and 2.
In the West, the sequence was first studied by Leonardo of Pisa, known as Fibonacci, in his Liber Abaci (1202)[3]. He considers the growth of an idealised (biologically unrealistic) rabbit population, assuming that:
* in the first month there is just one newly-born pair,
* new-born pairs become fertile from their second month on
* each month every fertile pair begets a new pair, and
* the rabbits never die
Let the population at month n be F(n). At this time, only rabbits who were alive at month n-2 are fertile and produce offspring, so F(n-2) pairs are added to the current population of F(n-1). Thus the total is F(n) = F(n-1) + F(n-2).[4]
In mathematics, a perfect number is defined as an integer which is the sum of its proper positive divisors, that is, the sum of the positive divisors not including the number. Equivalently, a perfect number is a number that is half the sum of all of its positive divisors, or s(n) = 2 n.
The first perfect number is 6, because 1, 2 and 3 are its proper positive divisors and 1 + 2 + 3 = 6. The next perfect number is 28 = 1 + 2 + 4 + 7 + 14. The next perfect numbers are 496 and 8128.
Even perfect numbers
Euclid discovered that the first four perfect numbers are generated by the formula 2n-1(2n - 1):
for n = 2: 21(22 - 1) = 6
for n = 3: 22(23 - 1) = 28
for n = 5: 24(25 - 1) = 496
for n = 7: 26(27 - 1) = 8128
Noticing that 2n - 1 is a prime number in each instance, Euclid proved that the formula 2n-1(2n - 1) gives an even perfect number whenever 2n - 1 is prime (Euclid, Prop. IX.36).
Ancient mathematicians made many assumptions about perfect numbers based on the four they knew. Most of the assumptions were wrong. One of these assumptions was that since 2, 3, 5, and 7 are precisely the first four primes, the fifth perfect number would be obtained when n = 11, the fifth prime. However, 211 - 1 = 2047 = 23 · 89 is not prime and therefore n = 11 does not yield a perfect number. Two other wrong assumptions were:
* The fifth perfect number would have five digits since the first four had 1, 2, 3, and 4 digits respectively.
* The perfect numbers would alternately end in 6 or 8.
The fifth perfect number (33550336 = 212(213 - 1)) has 8 digits, thus refuting the first assumption. For the second assumption, the fifth perfect number indeed ends with a 6. However, the sixth (8 589 869 056) also ends in a 6. It is straightforward to show the last digit of any even perfect number must be 6 or 8.
In order for 2n - 1 to be prime, it is necessary that n should be prime. Prime numbers of the form 2n - 1 are known as Mersenne primes, after the seventeenth-century monk Marin Mersenne, who studied number theory and perfect numbers.
Two millennia after Euclid, Euler proved that the formula 2n-1(2n - 1) will yield all the even perfect numbers. Thus, every Mersenne prime will yield a distinct even perfect number—there is a concrete one-to-one association between even perfect numbers and Mersenne primes. This result is often referred to as the "Euclid-Euler Theorem". As of December 2006 only 44 Mersenne primes are known, which means there are 44 perfect numbers known, the largest being 232,582,656 × (232,582,657 - 1) with 19,616,714 digits.
The first 39 even perfect numbers are 2n-1(2n - 1) for
n = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917 (sequence A000043 in OEIS)
The other 5 known are for n = 20996011, 24036583, 25964951, 30402457, 32582657. As of 2006 it is not known whether there are others between them.
It is still uncertain whether there are infinitely many Mersenne primes and perfect numbers. The search for new Mersenne primes is the goal of the GIMPS distributed computing project.
Since any even perfect number has the form 2n-1(2n - 1), it is a triangular number, and, like all triangular numbers, it is the sum of all natural numbers up to a certain point; in this case: 2n - 1. Furthermore, any even perfect number except the first one is the sum of the first 2(n-1)/2 odd cubes:
6 = 2^1(2^2-1) = 1+2+3, \,
28 = 2^2(2^3-1) = 1+2+3+4+5+6+7 = 1^3+3^3, \,
496 = 2^4(2^5-1) = 1+2+3+\cdots+29+30+31 = 1^3+3^3+5^3+7^3, \,
8128 = 2^6(2^7-1) = 1+2+3+\cdots+125+126+127 = 1^3+3^3+5^3+7^3+9^3+11^3+13^3+15^3. \,
It is unknown whether there are any odd perfect numbers. Various results have been obtained, but none that has helped to locate one or otherwise resolve the question of their existence. Carl Pomerance has presented a heuristic argument which suggests that no odd perfect numbers exist. [1] Also, it has been conjectured that there are no odd Ore's harmonic numbers. If true, this would imply that there are no odd perfect numbers.
Any odd perfect number N must be of the form 12m + 1 or 36m + 9 and satisfy the following conditions:
* N is of the form
N=q^{\alpha} p_1^{2e_1} \ldots p_k^{2e_k},
where q, p1, …, pk are distinct primes and in modulo 4 arithmetic q = a = 1 (Euler).
* N has either qa > 1020 or p_j^{2e_j} > 1020 for some j (Graeme Laurence Cohen 1987).
* The smallest prime factor of N is less than (2n + 6) / 3 where n is the number of distinct prime factors (so with k as used above, n = k + 1) (Grün 1952).
* The largest prime factor of N is greater than 108 (Takeshi Goto and Yasuo Ohno, 2006).
* The second largest prime factor is greater than 104, and the third largest prime factor is greater than 100 (Iannucci 1999, 2000).
* N has at least 75 prime factors; and at least 9 distinct prime factors. If 3 is not one of the factors of N, then N has at least 12 distinct prime factors. (Nielsen 2006; Kevin Hare 2005).
* N is less than 2^{4^{n}} (Nielsen 2003).
* N does not have e1=e2...=ek = 1 (modulo 3) (McDaniel 1970).
If N exists, it must be greater than 10300. A proof is expected for 10500 soon. See [2] for more information.
In case of e1 = e2 = ... = ek = ß in the factorization above, Tomohiro Yamada proved that there are only finitely many odd perfect numbers for any given ß. There are no odd perfect numbers when ß is equal to 1, 2, 3, 5, 6, 8, 11, 12, 17, 24 or 62 (Steuerwald, McDaniel, Kanold, Hagis, Cohen, Williams).[citation needed] There are no odd perfect numbers when ß is of the form 3k+1, from McDaniel's theorem.
Even perfect numbers have a very precise form; odd perfect numbers are rare, if indeed they do exist. There are a number of results on perfect numbers that are actually quite easy to prove but nevertheless superficially impressive; some of them also come under Richard Guy's Strong Law of Small Numbers:
* Stuyvaert: Every odd perfect number is the sum of two squares. (1896)
* Makowski: The only even perfect number of the form x3 + 1 is 28. (1962)
* By dividing the definition through by the perfect number N, the reciprocals of the factors of a perfect number N must add up to 2:
o For 6, we have 1 / 6 + 1 / 3 + 1 / 2 + 1 / 1 = 2;
o For 28, we have 1 / 28 + 1 / 14 + 1 / 7 + 1 / 4 + 1 / 2 + 1 / 1 = 2, etc.
* The number of divisors of a perfect number (whether even or odd) must be even, since N cannot be a perfect square.
o From these two results it follows that every perfect number is an Ore's harmonic number.
* Curtiss (1922) uses a greedy algorithm for Egyptian fractions to prove that a perfect number N must have a number of divisors at least proportional to lnlnN. A much stronger singly-logarithmic bound would follow from the nonexistence of odd perfect numbers and the known form of even perfect numbers.
The sum of proper divisors gives various other kinds of numbers. Numbers where the sum is less than the number itself are called deficient, and where it is greater than the number, abundant. These terms, together with perfect itself, come from Greek numerology. A pair of numbers which are the sum of each other's proper divisors are called amicable, and larger cycles of numbers are called sociable. A positive integer such that every smaller positive integer is a sum of distinct divisors of it is a practical number.
By definition, a perfect number is a fixed point of the restricted divisor function s(n) = s(n) - n, and the aliquot sequence associated with a perfect number is a constant sequence.
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