Amicable numbers Saud Dhaafi Mathematics Department, Western Oregon University May 11, 2021 1 / 36 • 1 is the first deficient number. • 6 is the first perfect number. • 12 is the first abundant number. • An amicable numbers are two different numbers m and n where σ(m)=σ(n) = m + n and m < n. • σ(n) is the sum of all positive divisors of a positive integer n including 1 and n itself. • A proper divisor is the function s(n) = σ(n) − n. Introduction • We know that all numbers are interesting. Zero is no amount while it is an even number. 2 / 36 • 6 is the first perfect number. • 12 is the first abundant number. • An amicable numbers are two different numbers m and n where σ(m)=σ(n) = m + n and m < n. • σ(n) is the sum of all positive divisors of a positive integer n including 1 and n itself. • A proper divisor is the function s(n) = σ(n) − n. Introduction • We know that all numbers are interesting. Zero is no amount while it is an even number. • 1 is the first deficient number. 3 / 36 • 12 is the first abundant number. • An amicable numbers are two different numbers m and n where σ(m)=σ(n) = m + n and m < n. • σ(n) is the sum of all positive divisors of a positive integer n including 1 and n itself. • A proper divisor is the function s(n) = σ(n) − n. Introduction • We know that all numbers are interesting. Zero is no amount while it is an even number. • 1 is the first deficient number. • 6 is the first perfect number. 4 / 36 • An amicable numbers are two different numbers m and n where σ(m)=σ(n) = m + n and m < n. • σ(n) is the sum of all positive divisors of a positive integer n including 1 and n itself. • A proper divisor is the function s(n) = σ(n) − n. Introduction • We know that all numbers are interesting. Zero is no amount while it is an even number. • 1 is the first deficient number. • 6 is the first perfect number. • 12 is the first abundant number. 5 / 36 • σ(n) is the sum of all positive divisors of a positive integer n including 1 and n itself. • A proper divisor is the function s(n) = σ(n) − n. Introduction • We know that all numbers are interesting. Zero is no amount while it is an even number. • 1 is the first deficient number. • 6 is the first perfect number. • 12 is the first abundant number. • An amicable numbers are two different numbers m and n where σ(m)=σ(n) = m + n and m < n. 6 / 36 • A proper divisor is the function s(n) = σ(n) − n. Introduction • We know that all numbers are interesting. Zero is no amount while it is an even number. • 1 is the first deficient number. • 6 is the first perfect number. • 12 is the first abundant number. • An amicable numbers are two different numbers m and n where σ(m)=σ(n) = m + n and m < n. • σ(n) is the sum of all positive divisors of a positive integer n including 1 and n itself. 7 / 36 Introduction • We know that all numbers are interesting. Zero is no amount while it is an even number. • 1 is the first deficient number. • 6 is the first perfect number. • 12 is the first abundant number. • An amicable numbers are two different numbers m and n where σ(m)=σ(n) = m + n and m < n. • σ(n) is the sum of all positive divisors of a positive integer n including 1 and n itself. • A proper divisor is the function s(n) = σ(n) − n. 8 / 36 Example Divisors of n n σ(n) s(n) = σ(n) − n σ(n) is (=; <; or >) 2n 6 1 + 2 + 3 + 6 = 12 6 = 12 • 15 1 + 3 + 5 + 15 = 24 9 < 30 18 1 + 2 + 3 + 6 + 9 + 18 = 39 21 > 36 Perfect, Deficient, Abundant Number • The number n is perfect if and only if σ(n) = 2n, deficient if and only if σ(n) < 2n, and abundant if and only if σ(n) > 2n. 9 / 36 Perfect, Deficient, Abundant Number • The number n is perfect if and only if σ(n) = 2n, deficient if and only if σ(n) < 2n, and abundant if and only if σ(n) > 2n. Example Divisors of n n σ(n) s(n) = σ(n) − n σ(n) is (=; <; or >) 2n 6 1 + 2 + 3 + 6 = 12 6 = 12 • 15 1 + 3 + 5 + 15 = 24 9 < 30 18 1 + 2 + 3 + 6 + 9 + 18 = 39 21 > 36 10 / 36 • (220; 284); (1184; 2924); (5020; 5564). • σ(220) = 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 + 220 = 504. • σ(284) = 1 + 2 + 4 + 71 + 142 + 284 = 504. • 220 + 284 = 504 = σ(220) = σ(284). • we see that a proper divisor of σ(200) = σ(284) which is 504 − 220 = 284 504 − 284 = 220. • The total of amicable pairs is unknown, but as of May 2021 the total of known amicable pairs is 1; 226; 910; 693. • The first amicable general formula discovered by Thabit Ibn Qurra. Amicable Numbers • An amicable numbers are two different numbers m and n where σ(m)=σ(n) = m + n and m < n. 11 / 36 • σ(220) = 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 + 220 = 504. • σ(284) = 1 + 2 + 4 + 71 + 142 + 284 = 504. • 220 + 284 = 504 = σ(220) = σ(284). • we see that a proper divisor of σ(200) = σ(284) which is 504 − 220 = 284 504 − 284 = 220. • The total of amicable pairs is unknown, but as of May 2021 the total of known amicable pairs is 1; 226; 910; 693. • The first amicable general formula discovered by Thabit Ibn Qurra. Amicable Numbers • An amicable numbers are two different numbers m and n where σ(m)=σ(n) = m + n and m < n. • (220; 284); (1184; 2924); (5020; 5564). 12 / 36 • σ(284) = 1 + 2 + 4 + 71 + 142 + 284 = 504. • 220 + 284 = 504 = σ(220) = σ(284). • we see that a proper divisor of σ(200) = σ(284) which is 504 − 220 = 284 504 − 284 = 220. • The total of amicable pairs is unknown, but as of May 2021 the total of known amicable pairs is 1; 226; 910; 693. • The first amicable general formula discovered by Thabit Ibn Qurra. Amicable Numbers • An amicable numbers are two different numbers m and n where σ(m)=σ(n) = m + n and m < n. • (220; 284); (1184; 2924); (5020; 5564). • σ(220) = 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 + 220 = 504. 13 / 36 • 220 + 284 = 504 = σ(220) = σ(284). • we see that a proper divisor of σ(200) = σ(284) which is 504 − 220 = 284 504 − 284 = 220. • The total of amicable pairs is unknown, but as of May 2021 the total of known amicable pairs is 1; 226; 910; 693. • The first amicable general formula discovered by Thabit Ibn Qurra. Amicable Numbers • An amicable numbers are two different numbers m and n where σ(m)=σ(n) = m + n and m < n. • (220; 284); (1184; 2924); (5020; 5564). • σ(220) = 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 + 220 = 504. • σ(284) = 1 + 2 + 4 + 71 + 142 + 284 = 504. 14 / 36 • we see that a proper divisor of σ(200) = σ(284) which is 504 − 220 = 284 504 − 284 = 220. • The total of amicable pairs is unknown, but as of May 2021 the total of known amicable pairs is 1; 226; 910; 693. • The first amicable general formula discovered by Thabit Ibn Qurra. Amicable Numbers • An amicable numbers are two different numbers m and n where σ(m)=σ(n) = m + n and m < n. • (220; 284); (1184; 2924); (5020; 5564). • σ(220) = 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 + 220 = 504. • σ(284) = 1 + 2 + 4 + 71 + 142 + 284 = 504. • 220 + 284 = 504 = σ(220) = σ(284). 15 / 36 • The total of amicable pairs is unknown, but as of May 2021 the total of known amicable pairs is 1; 226; 910; 693. • The first amicable general formula discovered by Thabit Ibn Qurra. Amicable Numbers • An amicable numbers are two different numbers m and n where σ(m)=σ(n) = m + n and m < n. • (220; 284); (1184; 2924); (5020; 5564). • σ(220) = 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 + 220 = 504. • σ(284) = 1 + 2 + 4 + 71 + 142 + 284 = 504. • 220 + 284 = 504 = σ(220) = σ(284). • we see that a proper divisor of σ(200) = σ(284) which is 504 − 220 = 284 504 − 284 = 220. 16 / 36 • The first amicable general formula discovered by Thabit Ibn Qurra. Amicable Numbers • An amicable numbers are two different numbers m and n where σ(m)=σ(n) = m + n and m < n. • (220; 284); (1184; 2924); (5020; 5564). • σ(220) = 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 + 220 = 504. • σ(284) = 1 + 2 + 4 + 71 + 142 + 284 = 504. • 220 + 284 = 504 = σ(220) = σ(284). • we see that a proper divisor of σ(200) = σ(284) which is 504 − 220 = 284 504 − 284 = 220. • The total of amicable pairs is unknown, but as of May 2021 the total of known amicable pairs is 1; 226; 910; 693. 17 / 36 Amicable Numbers • An amicable numbers are two different numbers m and n where σ(m)=σ(n) = m + n and m < n. • (220; 284); (1184; 2924); (5020; 5564).