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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with with permission permission of the of copyright the copyright owner. owner.Further reproduction Further reproduction prohibited without prohibited permission. without permission. KAMAL AL-DIN FARSI AND AMICABLE NUMBERS
by
Behzad Jalali
submitted to the
Faculty of the College of Arts and Sciences
of The American University
in Partial Fulfillment of
the Requirements for the Degree of
Master of Arts
in
Mathematics
Chair:
I-Lok Chang
Ali Enayat
Dean of the College of Arts and Sciences
DateD a t e 0 1995
-The American University Washington, D.C. 20016
£K£ IMS&ICiii UMIVEESITY LIBRABY
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: 1381752
UMI Microform 1381752 Copyright 1996, by UMI Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code.
UMI 300 North Zeeb Road Ann Arbor, MI 48103
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. KAMAL AL-DIN FARSI AND AMICABLE NUMBERS
BY
Behzad Jalali
ABSTRACT
The purpose of this paper is to introduce the work of Kamal AI-Din Farsi, a
thirteenth century Islamic (Iranian) mathematician. In order to verify the rule for finding
pairs of amicable numbers, set by Thabit Ibn Qurra, Kamal Al-Din wrote an article
consisting of twenty-five propositions. Kamal Al-Din’s article was translated from Arabic to
Farsi by Abulghassem Ghorbani in 1984 in a book called Farsi Nameh published by Homa
Publication in Tehran 1984. This thesis studies and translates Kamal Al-Din’s article on
amicable numbers from Farsi to English.
ii
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. TABLE OF CONTENTS
ABSTRACT ...... ii
LIST OF TABLES ...... iv
INTRODUCTION...... 1
Chapter
1 . PRELIMINARY NUMBER THEORETIC RESULTS...... 6
2. DIVISORS ...... 13
3. AMICABLE NUMBERS...... 20
4. CONCLUSION...... 34
BIBLIOGRAPHY...... 36
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LIST OF TABLES
Tables Page
1. Table of Summations ...... 14
2. Table of D ivisors ...... 16
iv
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. INTRODUCTION
Two numbers are called amicable if each is the sum of proper divisors of the
other. If we let pair m and n are amicableif tr(m) - m = n and or a(m) = m + n = a(n). Amicable numbers have been important in magic and astrology, in casting horoscopes, and in making talismans. The Greek believed these numbers had a particular influence in establishing friendship between two individuals. In The History of Mathematics Burton states, "The philosopher lamblichus of Chalcis (ca. A.D. 250- A.D. 330) ascribed a knowledge of the pair 220 and 284 to the Pythagoreans. He wrote: They (the Pythagoreans) called certain numbers amicable numbers, adopting virtues and social qualities to numbers, as 284 and 220, for the parts of each have the power to generate the other, according to the rule of friendship, as Pythagoras affirmed. When asked what is a friend, he replied "another I," which is shown in these numbers. However, Pythagoras may not be the only one to know of amicable numbers. Bible commentators point to Jacob’s gift of 220 goats to Esau on their reunion - a friendly gift? (Burton 1991, 462) It was not until Thabit Ibn Qurra (826-900), the brilliant Islamic mathematician, astronomer and physician, that the first rule for constructing amicable (Wells 1987, 220) numbers was presented. He described in his book, On the 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 Determination of Amicable Numbers. Euclid’s rule for perfect numbers, means of constructing abundant and deficient numbers, and the first rule for constructing amicable numbers, from which he deduced Pythagoras’ pair. Mehdi Nakosteen in his book, History of Islamic Origin of Western Education (1964) writes: Thabit Ibn Qurra Al-Harrani (826-900) of Harran made many translations of Greek works on medicine and mathematics, including the works of Apollonius, Archimedes, Euclid, Theodosius, Ptolomy, Galen, and Eutocius. The founder of a school of translators, he published solar observations and works on mathematics and anatomical and medical materials, and on astronomy (Nakhosteen 1964, 25). Thabit’s rule for finding amicable pairs of numbers is as follows: Theorem: If the three numbers p = 3 x2" -1, q = 3 x2"_l- l and r = 9 x2 2 '* - 1 -1 are all primes then, the two numbers Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3 M = 2mxpq and N = 2"xr are amicable. The second pair of amicable numbers was found around the beginning of the 14th century A.D. by another Islamic mathematician and scholar called Kamal Al-Din Farsi. Hassan son of Ali son of Hassan Kamal Al-Din Farsi, mathematician and physicist, was born in late 13th century A.D. in Fars, central Iran. During his youth he traveled extensively to meet the great scholars of his time. His teacher Qutb Al- Din Shirazi spoke of him very highly. His most important works were in mathematics and optics. Nakosteen writes: The Ketab Al-Manazer of Ibn Al-Haitham, dealing in part with optics, was translated into Latin by Gerard of Cremona, and the book became a basis for both Muslim and Latin opticians. An elaborate commentary on Ketab Al-Manazer was written at the beginning of the fourteenth century by Kamal Al-Din Al-Farsi, a disciple of Qutb Al-Din Al- Shirazi, the author of Nihayat Al-Idrak (The End of Thought) in which he gives an account of the rainbow (Nakhosteen 1964, 191). Kamal Al-Din Farsi died in the year 1318 A.D. in the city of Tabriz in Iran. The second pair of amicable numbers, 17296 and 18416, were found according to Thabit’s rule for n = 4. This pair was then rediscovered in 1636 by Fermat, who also rediscovered Thabit’s rule, as did Descartes, who produced a third pair, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4 9,363,584 and 9,437,056, two years later. The third pair comes from Thabit’s formula for n = 7. In 1747, Euler added to the three pairs of amicable numbers known to Fermat, bringing the list up to thirty pairs; later he increased this to more than sixty. His methods are still the basis for present-day exploration. Thabit’s rule for finding amicable numbers was known to and discussed among Islamic mathematicians. Qutb Al-Din Shirazi asked his disciple Kamal Al-Din Farsi to study Thabit’s work on amicable numbers and to verify the validity of his rule for finding such numbers. The result was an article Kamal Al-Din wrote stating and proving around twenty-five theorems and problems leading to proof of Thabit’s rule. We shall examine Kamal Al-Din’s paper and compare his theorems and problems with other number theoretic discoveries of mathematicians of Europe. For example, his Proposition 17 is one that Descartes stated and proved in 1638. Theorem (Kamal Al-Din) Let S(n)=Sum of proper divisors of n (Miller and Whalen 1995, 256). If the prime p does not divide the composite a then: S(ap) = S(a) • p + S(a) + a Kamal Al-Din for the first time stated and proved the above theorem. About 320 years after Al-Din’s death, Descartes stated the same theorem as follows: Theorem (Descartes) If p is a prime and b = S(a) where a is not prime, then Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. S(ap) = bp + a + b Note that the condition that p must not divide a is not stated in Descartes theorem. The purpose of this paper is to explore the number theoretic work of Kamal Al-Din Farsi, with special attention to his seminal work on amicable numbers. The source is his original work (in Farsi). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER I Preliminary Number Theoretic Results It was mentioned in the introduction of this paper that Thabit Ibn Qurra was the first Islamic scholar to study amicable numbers in depth. In his article on amicable numbers he states that two types of numbers and their constructions were important to Pythagoras and his followers: perfect and amicable numbers. Then he gives a definition of these numbers and says that Nicomachus of Gerasa has explained an algorithm to find perfect numbers but has not proved that his algorithm works. On the other hand, he notes that Euclid not only has shown the way to construct perfect numbers, but he has also proved the validity of his construction. And then Thabit says "But regarding amicable numbers I did not find anything in the works of these two". And then he continues his article about finding amicable numbers. Thabit states and proves ten theorems and propositions, where his 10th theorem is an algorithm to find amicable numbers, but there is no proof for this algorithm. It was not until Kamal Al-Din Farsi that Thabit’s rule for finding amicable numbers was proved. Both Thabit and Kamal Al-Din were well versed in Euclid’s number theory works and in their articles proofs of certain theorems and propositions are not stated, but rather related to Euclid’s works. It is also noted that Kamal Al- Din did not have access to Thabit’s article on amicable numbers. He only knew the way to construct them and then he wrote his article to verify the way. 6 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 7 Kamal Al-Din’s Article on Amicable Numbers Proposition 1. (Fundamental Theorem of Arithmetic) Every composite number is inevitably factored into a product of a finite number of prime numbers. It should be mentioned that number here has the same definition as Euclid’s: A number is an entity which is composed of units. (Positive Integer). Proof: Assume a is a composite number. Then according to the Proposition 29 of Book VII of Euclid (Euclid 1956, 2:330) a is divisible by a prime number b, so a—be where c also divides a. If c is prime, the theorem is proved and if c is not prime, then c—de and so a=bde. If e is prime the theorem is proved and if not we continue until we reach the product of two prime numbers. In this case a has been decomposed into the product of finitely many prime numbers. If we do not eventually reach two prime numbers, a must be equal to the product of infinitely many numbers and this is not possible. Proposition 2. If three numbers a,b and c are given, then the ratio of the first to third is the same as the ratio of the first to second multiplied by the ratio of the second to third or a _ c b e Proof: Kamal Al-Din has proved this theorem using Proposition 5 of Book VIII Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 8 (Euclid 1956, 2:354) and Proposition 18 of Book VII (Euclid 1956, 2:318). Let ba=d and bc=e Now according to Proposition 5 of Book VIII of Euclid we have: d ba B.x— e be b e Now let h = b2 According to Proposition 18 of Book VII of Euclid -v d _ ba _ a h b2 = b 2) * = * ! = * e be c from 1 and 2 a _ d _ ba c e be Proposition 3. The inverse of every composite number is equal to the product of inverses of each of its prime factors. In another words, if a and b are prime then Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Proposition 4. Two composite numbers that have the same prime factors and corresponding primes to the same exponents are equal. Proposition 5. If two composite numbers are not equal, then they do not have exactly the same prime factors. Proposition 6 . If a composite number is factored into its n prime factors, the product of any 2 or 3 or n-1 factors of the number are also divisors of the number. For example, if N = ab3 = axbxbxb then ab/ibz,b2,b3 divide N Proposition 7. If a does not divide b then a* and higher powers of a do not divide a*b, and a 3 and higher powers of a do not divide cfb and a4 and higher powers of a do not divide a*b and so on. To prove this theorem Kamal Al-Din used Proposition 18 of book VII of Euclid (Euclid 1956, 2:318). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 10 Proposition 8 . Let the composite N be prime factorized. i) If a is represented once as a prime factor, then a2, a3, ...a" do not divide N. ii) If a is represented twice in the factorization, then a and a2 divide N but no higher power of a divides N. To prove this Proposition, Kamal Al-Din has used Proposition 24 and 25 of Book VII of Euclid (Euclid 1956, 2:325-26). iii) If a is represented 3 times in the factorization, then a, a2,and a3 divide N and no higher power of a, and so on. Proposition 9. Let the composite N be prime factorized into its n prime numbers; then the divisors of N are 1, its prime factors, the product of the primes 2 at a time or three at a time or (n- 1 ) at the time. For example: if V=5x7 3 = 5x7x7x7 then divisors of N are 1, 5, 7, 5x7, 7x7, 5x7x7, 7x7x7, and there is no other proper divisor of N. Proposition 10. Every even number is composite except 2. Proof: Every even number is divisible by 1/2 of itself and also 2. Proposition 11. Every odd number is either a single-digit odd number or there is a odd number in its "one’s place." Proof: Every number is one of the four cases. i) A single digit number ii) One digit followed by zeros. (A "Solitary" Number). iii) Several digits where the "one’s place value" is not zero iv) Several digits and the one’s place is zero Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. II Proof: If the odd number is of the 1st or 3rd type then the theorem is proved and it is obvious that an odd number is not of 2nd or 4th type (according to Proposition 21 of Book IX of Euclid) (Euclid 1956, 2:413). Proposition 12. If a number ends with one or more zeros and the last digit before the zeros is 5, then that number is divisible by 5. Proposition 13. If a number is divisible by 5, then its last digit is zero or five. Deaf Numbers Proposition 14. The smallest number divisible by any one from a finite set of numbers is equal to the square of the smallest number from the set. For example, if a, b, c are numbers such that a divisible by any one of a,b,or c is a2. Definition: A number is called "deaf" if it is not divisible by 2,3,4,5, 6 ,7,8 ,9, and 10. Or a number is "dear if the result of a mathematical operation performed on that number can only be approximated. For example, V5 is a deaf square root because it can only be approximated by a ratio. The author has deduced from Proposition 14 that the smallest composite "deaf" number is 1 2 1 because the smallest "deaf" number is 1 1 . Prime Factorization Problem 1. We want to know if a number is prime or composite and if it is composite, how to factor it into prime numbers. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 12 Here the author, by using the previous proposition of his article and Proposition 21 of Book DC of Euclid, argues as follows (Euclid 1956, 2:413). 1) If the number a is even and not equal to 2, then it is not prime. 2) If a is odd, then it consists of 1 or several digits. If it is a one digit number it is prime except if it is equal to 9. If a has more than one digit and its last digit is 5, then it is not prime, but if its last digit is 1,3,7, or 9 then it may be prime. Then the author says if the number is "deaf" and smaller than 121 then it is prime, but if it is larger than 1 2 1 then we would divide it by 1 2 1 if it has no remainder or its remainder is divisible by 1 1 then it is not prime, if the number is not divisible by 1 2 1 or its remainder upon dividing by 1 2 1 is not divisible by 11 then it may be prime. In this case, we divide the number by the square of the second "deaf" number, 13. Procedure to find prime factors of a composite number: In conclusion of this argument Kamal Al-Din says: In order to discover divisors (prime factors) of a number, first we divide the number by one of its divisors we learned how to discover by Problem 1. If the quotient and divisor are both prime then we are done; otherwise we continue with the process until the quotient and the divisor are both prime. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter n Divisors At this point the author introduces some definitions under the heading of "SADR", meaning "Beginning, proemium or exordium of a letter" to use later in presenting new propositions. Summations Definition: FIRST SUMMATIONS A sequence where the n* term is the sum of the first n+1 natural numbers. Definition: SECOND SUMMATIONS A sequence where the n* term is the sum of the first n terms of "First summations plus I." Third and fourth summations are defined as above; for example, third summations are: 5,15,35,70,126,... Kamal Al-Din wrote a table consisting of ten of these summations and called it "table of summations." He made this table available for students to use and follow the pattern if other summations are needed. The following is a partial listing of the table of summations. 13 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 14 Table ITable of Summations 1st Term 2nd Term 3rd Term 1st Summations 3 6 1 0 2nd Summations 4 1 0 2 0 3rd Summations 5 15 35 Notice the entry at the m* row and n* column is (I) where a=n+m+l and /3=m+l. The Number of Divisors Proposition 16. How to obtain all proper divisors of a composite number. Case 1: All prime factors of the composite number N are equal: N =ar. In this case divisors of N are powers of a like a* such that 0 < j < r. Case 2: All prime factors of N are different and there are not many of them. For example, N=bcde. Then the author says: First we multiply the factors two at a time; he calls them "two at the time factors." To obtain divisors which are formed by multiplying prime factors three at the time, we exclude each prime factor Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 15 and multiply the remaining three factors. And then of course we must add to these the prime factors themselves. Case 3: All prime factors of N are different but there are many of them. Then the authors says, if N has n different factors, in order to obtain the number of its divisors which are formed by multiplying its prime factors r at the time r < n, we must refer to "table of summations" and the desired number is at the (r-l ) " 1 row and (n-r)* column. For example, if the composite number N has eight different prime factors then the number of its divisors that are obtained by multiplying its prime factors three at the time can be found in "table of summations" row 2 and column 5. Case 4: Some of the prime factors are equal. In this case the author says that this case is like case three. We assume all prime factors are different but upon calculating divisors we ignore reoccurrences. For example in order to obtain all divisors of N =a3bc which are the product of two prime factors at a time we write a3 = a Ia2 a3, where a 1 =a 2 =a3; according to case 3; these divisors are be, 3jC, 8 3 b , 8 3 0 ,8 2 b, ajC, ajb, 8 2 8 3 ,8 4 8 3 , 3 ^ 8 2 but ata 2 = 8 4 8 3 = 8 3 8 3 = a2 a,b=a 2 b = a 3 b= ab a4c=a2c=a3c=ac Therefore the number of distinct such divisors is four: bc,ac,ab,a2. Table of Divisors At this point Kamal Al-Din introduces a table that he calls "table of divisors" in which he gives examples of composite numbers which have from two to ten prime Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 16 factors with their number of divisors; he uses "ABJAD " 1 letters to represent prime factors. He also writes letters next to one another to represent products; however he does not use exponents. For example, for a3 he writes aaa.The following table is a partial listing of the author’s table: Table 2:—Table of Divisors N Number of Divisors Prime Factors 32 5 AAAAA 48 9 AAAAB 72 1 1 AAABB 1 2 0 15 AAABJ 180 17 AABBJ 420 23 AABJD In this table Kamal Al-Din has named numbers by their number of prime factors. The last example is the "9 factor number" AAAABBBJJ=2 4 x3 3 x52= 10,800, which has 59 proper divisors. lA rearrangement of Arabic alphabet letters with specific numerical values. Used in esoteric sciences such as "science of numbers", alchemy, etc. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 17 Kamal Al-Din’s Rule Proposition 17. If a composite a is multiplied by a prime p such that p does not divide a, then sum of the proper divisors of ap is equal to the sum of the proper divisors of a multiplied by p plus the sum of the proper divisors of a plus a. Or S(ap) =S(a)p+S(a) +a Kamal Al-Din was the first to state and prove the above proposition. In 1638, 320 years after the death of Kamal Al-Din, Descartes stated the following theorem: If a is composite and pa prime then, S(ap) =S(a)p+S(a)+a Note that it is not stated that p must not divide a. Kamal Al-Din has proved the above proposition with extensive explanation and has also considered the case in which p is a divisor of a, which we shall discuss after the following example. Now if N = a“b'V , then number of divisors of N is the number of terms of the expansion of (l+a+ ... +a“)(l+b+ ... +b**)(l+c+ ... +cT) which is (a+ l)0 8 + l)(7 + l). Therefore the number of proper divisors of N is (a+l)G 8 + l)( 7 + l)-l. Also the sum of the divisors of N is (l+a+ ... +a“)(l+b+ ... +b's)(l+c+ ... +cT), which is the same as Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 18 a**1-! b*+l- l cv+1- l a - 1 b- 1 c - 1 Therefore the sum of the proper divisors of N is a a*l- l b* * 1 - 1 c Y+1-l S(N) = ------N a - 1 6 - 1 c - 1 Example 1. Let a=30=2x3x5 and p = ll; then ap=2«3#5*l 1=330. Therefore S(a)=(2+1)(3 +1)(5+1)-30=42. According to Kamal Al-Din’s formula, S(ap) =42x11 +42+30=534, which is the same as S(ap) = (2 + 1)(3 + 1)(5+ 1)(11 + 1)-330=534. Example 2. Let a=147=3x72 and p=13 then ap=l911=3x72xl3 and 5(a) = (3+l)^i-147 = 81 7-1 According to Kamal Al-Din’s rule, S(ap)=81xl3+81 + 147= 1281, which is equal to S(ap): After proving Proposition 17 Kamal Al-Din discusses the case in which p is Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. one of the prime factors of a, but does not prove it. He remarks that he would not extend this matter any further because it does not concern his main topic. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter IQ Amicable Numbers Proposition 17A. If p is a prime factor of a, then S(ap) = (sum of the divisors of a which are multiples of p)-p+S(a)+a Example: Let a =30 =2x3x5 and p=5; then ap=150=2x3x52 and S (a)= l+ 2+ 3+ 5+ 6+ 10+ 15= 42 The sum of the divisors of a which are multiples of p is 5+10+15=30 Then S(ap) =30x5+42+30=222, which is the same as 22- l 32- l 55-l S(ap) = -— —--150 = 222 2-1 3-1 5-1 In general, if a= p“»b then S(ap) = (sum of the proper divisors of a which are multiples of p“)*p+S(a)+a Example. Let a =525=3x5V7 and p=5 ap=2625=3x53x7 The proper divisors of a are 1,3,5,15,25,75,7,21,35,105,175 and S(a)=467. The sum of the divisors of a which are multiples of 5 2 is 25 +75 + 175 =275. 20 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 21 According to Kamal Al-Din’s rule, S(ap) =275x5+467+525 =2367, which is the same as 32- l 54-1 72-l S(2625) = -— —--2625 = 2367 3-1 5-1 7-1 Proposition 18. If the two composite numbers a and b are multiplied together, the sum of the proper divisors of the product ab is equal to the product of b and the sum of the proper divisors of a, plus the product of a and the sum of the proper divisors of b plus the product of the sums of theproper divisors of a and b. This holds when the ratio of any divisor of a to any divisor of b is not equal to the ratio of any other divisor of a to any divisor of b (a and b are relatively prime) (case 1). But if such ratios are equal, we must subtract from the above result the cross product of all such ratios (case 2). In other words in case 1 where a and b are relatively prime; S(ab)=S(a)b+S(b)a+S(a)+S(b). (I) In 1638 Descartes stated and proved the rule for determining the sum of all divisors of the product of two numbers. If a(a) =sum of all divisors of a, then cr(ab)=cr(a)* Descartes has stated that for the above rule a and b must be relatively prime. However, he has not considered the case in which a and b are not relatively prime. Example 1. Let a =30=2x3x5 and b=91=7xl3. Clearly a and b are relatively prime. S(a)=(2+l)(3 + l)(5 + l)-30 =42 and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 22 S(b)=(7+l)(13+l)-91=21. According to Kamal Al-Din’s rule, S(ab) =42x91 +21x30+42x21 =5334, which is the same as S(ab)=(2+ 1)(3 + 1)(5+ 1)(7+1)(13 + 1)-2730 =5334. Example 2. The case (case 2) in which a and b are not relatively prime. Then S(ab)=S(a)*b+S(b)*a+S(a)S(b)-(sum of cross products of equal ratios of any divisors of a to any divisors of b) Let a = 6 =2x3 and b=15=3x5. Then S(a)=1 +2+3=6 S(b) = l+ 3 + 5 = 9 The divisors of a = 6 are 1,2,3,6 , and the divisors of b=15 are 1,3,5,15. Now among these numbers the following properties exist: 2 = A 1 = 2 1 = 2. 1 = 2 5 15’ 1 3 ’ 5 15’ 1 3 and the cross products are 2x15=30, 2x3=6, 1x15 = 15, 1x3=3 where 30+6+15+3=54. According to Kamal Al-Din’s rule, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 23 S(ab) =6x15+9x6+6x9-54=144, which is the same as S(ab) = — .— .— -90 = 144 2-1 3-1 5-1 Example 3. Let a = 15=3x5 and b=35 =5x7. ab=3x52x7=525 S(a) = l+3+5=9 S(b) = 1+5+7=13. Divisors of 15 are 1,3,5,15 Divisors of 35 are 1,5,7,35 Among these divisors the following properties exist: 2 = il 1 = 11 1 = _L 1=1 7 35’ 1 5 ’ 7 35* 1 5 Notice that the numerators are divisors of 15 and that the denominators are divisors of 35. Now the cross products are: 3x35 = 105, 3x5 = 15, 1x35=35 and 1x5=5 and their sum is equal to 105+15 +35 +5 = 160, and S(ab) = 9x35 + 13x15+9x13-160 = 467. This is the same as: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 24 32- l 53- l 72- l S(ab) = - —- .- —- .- — --525 = 467 3-1 5-1 7-1 Kamal Al-Din has also taken in account the case in which one of the composite numbers divides the other. Proposition 19. If the two numbers a and b are not the smallest numbers having the ratio equal to a b then they have a common divisor. Proof: If we consider all pairs of numbers whose ratios are the same and let c and d be the smallest pair and ab any other pair, then according to Proposition 20 of the VII book of Euclid we have (Euclid 1956, 2:320): c = —.a1 ^ a = 1 —.b * n n where n is a natural number, then a=nc, b=nd, where n is a common divisor of a and b. Proposition 20. Consider the two composite numbers a and b. If the ratio of two of the divisors of a is equal to the ratio of two of the divisors of b, then some of the divisors of a are equal to some of the divisors of b. For example: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 25 let a = 1 2 , b=18 then Divisors of a are: 1,2,3,4,6,12 Divisors of b are: 1,2,3,6,9,18 notice, _12 _ JL8 4 6 and some of the divisors of a are equal to some of the divisors of b, namely 2,3, and 6 are common divisors of a and b. Proposition 21 (Converse of Proposition 20). If the composites of a and b have a common divisor, the ratio of two of the divisors of a is equal to the ratio of two of the divisors of b. Proposition 22. If 1 is subtracted from any term of the geometric progression: Li1 2 ^y 22 * **•••* 2" y then the sum of the previous terms is obtained: 2 B - 1 = l+ 2 +2 2+ . . .+2 n _ 1 . Kamal Al-Din has proved this identity using Proposition 13 of book IX of Euclid (Euclid 1956, 2:399). Proposition 23. If we take one of the powers of 2, for example 2n, and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. determine the four following odd numbers: 1" odd number = b = 3x2"_1-l 2nd odd number = c = 3x2 *-1 3rd odd number = d =bxc = (3x2n- 1 -l)(3x2"-l) = 9x2 2 ,*" 1 -3x2,,"1 -3x2n+l 4rt odd number = e = b+c+d = 9x22n~l- l such that b,c and e are prime, then 2n.e-2n.d = e-(2x2 n- l) . ( 1 ) Proof: Let a =2“; then b = a+ 1- 1 (2 ) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 27 e = 3 a -l (3) d = be = (— -l)(3a-l) = 4a 2 +— + l-4 a -— (4) 2 2 2 „ 2 e = b+c+d = 4a2+— -1 (5) 2 a(c-d)=a(4c+--2) = 4a 2 +— -2a (6 ) 2 2 Adding (2a-l) to both sides of (6), we obtain a(e-d)+(2a-l) = 4a2+——1 (7) 2 Equating the left hand sides of (5) and (7) yields a(e-d)+(2a-l) = e ae-ad = e-( 2 a -l), Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 28 or 2Hxe-2Hxd = «-(2 x2 " -l) . Proposition 24. a=2“ and d=bxc=(3x2“' 1 -l)(3x2n-l) have no common divisors, except 1 . Proof: The divisors of a=2“ except 1 are 2 21, . . . 2 B_l which are all even, and the divisors of d are b and c, which are prime and odd. Proposition 25. If a = 3x2"_1-l b = 3x2"-l c = 9x22n~l- l are all prime, then m = 2 Dxab and n = 2 nc are amicable. Proof: We must show that the sum of the proper divisors of m is equal to n and that the sum of the proper divisors of n is equal to m. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 29 o(m)=m+n=o(ri) Notice according to equation (1) of Proposition 23, n-m = c-(2**l-l) . (1) Now we show that the sum of proper divisors of n is equal to m. The composite number 2“ is multiplied by c to produce m. According to Proposition 17 we have: S(N) =S(2").c +S(2K) +2", (2) provided the prime number c does not divide 2", whichis clearly the case. Now the sum of the proper divisors of 2 “ is 2 “-l; therefore S(2")+2n = 2"-1+2" = 2n+1-l (3) Now (2) can be written as: S(n) = (2"-1)(9x22*_1-1)+(2b+1-1) , (4) but (2"-1)(9x22,i_i-T) = 2n(9X22"'1 -1)-9x22n"1 +l=n-c; Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 30 therefore (4) can be written as: S(n) = n-c+(2H*l- l) (5) and ( 1 ) can be written as: m = n-c+(2 "+1 -l) (6 ) From (5) and ( 6 ) we obtain S(n) = m. Now we must show that S(m) = n. Since the composite number 2“ is multiplied by ab to obtain m, then according to Proposition 18: S(m) = S(2K).ab+S(ab)+2n+S(2n) (7) providing no pair of divisors of 2 “ is proportional to any pair of divisors of ab. According to Propositions 24, 20 and 21 this condition holds. Let h= 2“. Then Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 31 a = 3x2"-i- l = - h -1, 2 b = 3x2*-l = 3h-l. Then let ab = (—A-l)(3/t-l) = 2 2 2 Now we have c = 9x2"’l-l = - h 2-1 2 n = 2".c = /r(—A2-l) 2 m = 2nab = h(— -—1) + 2 2 S(2") = 2n- l = h -1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 32 and S(ab) = a+b+l = y - 1 Therefore equation (7) can be written as 9h2 9h 9 h 9h3 S(m) = (A -l)(^ -}£+l)+(^-l)(2A-l) = Z L - k 2 2 2 2 9h2 9^12h A(-^--l)=2"(—^ — 1) 2 2 =2n(9x22n'l - l )=n. At this point Kamal Al-Din under the heading "Conclusion of the Article" says, "Now that I am free from this task; I shall explain with a few examples." Example 1. Take one of the powers of 2, for example 4=22, and construct the four odd numbers according to Proposition 23: n = 2 a=3x22'I-l=5 b=3x22-l = ll Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 33 c=axb=55 d=a+b+c=5+l 1+55=71 Because a,b, and d are prime, m =22xc =4x55=220 and n=2*xd=4x71=284 are amicable, Kamal Al-Din verifies the amicability of these two numbers using Propositions 17 and 18. Example 2. Take 8=23 and let a=3x22-l = ll b=3x23-l=23 c=axb=253 d=a+b+c=287 Since d is not prime, we can not construct a pair of amicable numbers using 23. Example 3. Take 16=24 let a=3x23-l =23 b=3x24-l=47 c=axb=1081 d=a+b+c=1151 Since a,b, and d are prime, we have m = 16x1081 = 17296 and m = 16x1081 = 17296 and n= 16x1151 = 18416 are amicable. Kamal Al-Din also verifies the result. The pair 17296 and 18416 were first discovered in Europe by Fermat in 1638, 318 years after the death of Kamal Al-Din. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CONCLUSION Kamal Al-Din Farsi has only been recently discovered as a mathematician. His most important mathematical work so far studied is his article on amicable numbers. He also has articles on optics. It is evident in this paper that Kamal Al- Din had extensive knowledge of mathematics. It is probable that he also taught his knowledge of mathematics to students. He mentions this as he constructs his table of "summations" and table of "divisors" for the use of "students." A hand written copy of his book by one of his students is in Sepah Salar School Library in Tehran. Kamal Al-Din was well versed in Euclid’s works, which during the invasion of Europe by Muslim Arabs were translated from Greek to Arabic. However, Iranians were aware of the existence of Greek mathematics long before the advent of Islam. It is believed that after the invasion of Iran by Alexander of Macedonia, several thousand camel and mule loads of wealth, including which books from the libraries of Achamenians, were taken by Alexander. In addition, there were always Greek scholars residing in the courts of the Achamenian kings. Kamal Al-Din’s work is also important because of statements of the Fundamental Theorem of Arithmetic. Ahmet G. Agargun and Colin R. Fletcher state in their article "Al-Farsi and the Fundamental Theorem of Arithmetic," published in Historia Mathematica 21(1994) 102-173: 34 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 35 Proposition 1. Each composite [number] can necessarily be decomposed into a Finite number of prime factors of which it is the product. Let a be a composite number; since it is composite it is necessarily measured by a prime from Book VII.31 of the Elements. Let this [prime] be b, and let it [i.e., b\ measure it [i.e., a] by c. If c is a prime then it is shown that it [i.e., a] is made up by multiplying prime b and prime c. If it [i.e., c] is composite then let it be measured by a prime d according to the number e [i.e., c=de\. If e is prime then it is clear that a is made up by multiplying the prime numbers b, d and e. Otherwise we perform our operation until the composite factor is in the end decomposed into two prime factors. Then a is made up from the previous primes together with those two primes. If it never can be decomposed into two prime factors, then it would necessarily follow that the finite would be made up from an infinite product of numbers, which is absurd. And that is what we wanted. This is the first known statement and the first known proof of the existence of a prime decomposition of a given composite number. The Euclidean result quoted states that ’any composite number is measured by some prime number’ [2,332]. This is the first step on the road to proving al-Farisi’s Proposition 1, but it is not the existence theorem itself. No amount of juggling with words or economizing with the truth can turn the statement of VII.31 into Proposition 1. Note that al-Farisi, in the final stage, uses an argument by contradiction. Kamal Al-Din Farsi deserves his own place within the history of mathematics as a thirteenth century mathematician. This may open the door to further study and research regarding Kamal Al-Din’s other works, his mathematical training, his teachers, and his contemporaries. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. BIBLIOGRAPHY Agargun, Ahmet G. and Colin R. Fletcher. "al-Farisi and the Fundamental Theorem of Arithmetic." Historia Mathematica 21 (1994) : 162-73. Boyer, Carl B. A History of Mathematics. New York: John Wiley and Sons, Inc., 1968. Burton, David M. The History of Mathematics and Introduction. 2nd ed. Dubuque, IA: WCB, 1991. Euclid. The Thirteen Books of The Elements. Translated by Sir Thomas L. Heath. New York, N.Y., 1956. Ghorbani, A. Farsi Namah. Tehran: Homa Publication, 1984. Miller, Gordon L and Mary T. Whalen. " Multiply Abundant Numbers." School Science and Mathematics 5 (May 1995) : 256-59. Nakosteen, M. History of Islamic Origin of Western Education. Boulder, Colorado: University of Colorado Press, 1964. Wells, David. The Penguin Dictionary of Curious and Interesting Numbers. London: Penguin Group, 1987. 36 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.