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). • 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.

18 / 36 Thabit Ibn Qurra Theorem

Suppose for some integer n > 1, the numbers

a = 3 · 2n − 1 b = 3 · 2n−1 − 1 c = 9 · 22n−1 − 1

are all prime. Then the pair (2nab, 2nc) is amicable. Further, 2nab is abundant while 2nc is deficient. Notice that: b < a < c and a is almost double b while c is more than a · b.

19 / 36 • Find values for a,b and c which must prime. • 2nab is abundant while 2nc is deficient. Further, 2nab is the sum of the proper divisors of 2nc and at the same time 2nc is the sum of the proper divisors of 2nab

Steps

There are just three steps to follow to find an amicable number. • Choose a value for n where n > 1.

20 / 36 • 2nab is abundant while 2nc is deficient. Further, 2nab is the sum of the proper divisors of 2nc and at the same time 2nc is the sum of the proper divisors of 2nab

Steps

There are just three steps to follow to find an amicable number. • Choose a value for n where n > 1. • Find values for a,b and c which must prime.

21 / 36 Steps

There are just three steps to follow to find an amicable number. • Choose a value for n where n > 1. • Find values for a,b and c which must prime. • 2nab is abundant while 2nc is deficient. Further, 2nab is the sum of the proper divisors of 2nc and at the same time 2nc is the sum of the proper divisors of 2nab

22 / 36 Example

Example let n = 2 to 4 and

a = 3 · 2n − 1 b = 3 · 2n−1 − 1 c = 9 · 22n−1 − 1

n a b c prime not prime 2nab 2nc

2 11 5 71 all - 220 284

3 23 11 287 23,11 287 - -

4 47 23 1151 all - 17296 18416

(220, 284) and (17292, 18416) is amicable number by Thabit Ibn Qurra Method. 23 / 36 Euler Theorem

• Choose a value for a where a = gcd(m, n) • b a Using c = 2a−σ(a) , to find values for b and c. • Using b2 = (cx − b)(cy − b), to find values for (cx − b) and (cy − b), which will then be used to find values for x and y. • Finally, if p = x − 1, q = y − 1, and r = xy − 1 are all prime, then M = apq and N = ar are amicable numbers.

24 / 36 (x − 4)(y − 4) = 16 x − 4 y − 4 x y p = x − 1 q = y − 1 r = xy − 1 ••• 16 1 20 5 19 4 99 8 2 12 6 11 5 71 4 4 8 8 7 7 63 •

M = apq = 4 · 11 · 5 = 220 N = ar = 4 · 71 = 284 Thus, (220, 284)is amicable number by Euler method.

Example

Example Let a = 4 so

b a 4 4 4 4 = = = = = c 2a − σ(a) 2(4) − σ(4) 8 − (1 + 2 + 4) 8 − 7 1

so b = 4 and c = 1. Thus, from (cx − b)(cy − b) = b2 we got that:

25 / 36 x − 4 y − 4 x y p = x − 1 q = y − 1 r = xy − 1 ••• 16 1 20 5 19 4 99 8 2 12 6 11 5 71 4 4 8 8 7 7 63 •

M = apq = 4 · 11 · 5 = 220 N = ar = 4 · 71 = 284 Thus, (220, 284)is amicable number by Euler method.

Example

Example Let a = 4 so

b a 4 4 4 4 = = = = = c 2a − σ(a) 2(4) − σ(4) 8 − (1 + 2 + 4) 8 − 7 1

so b = 4 and c = 1. Thus, from (cx − b)(cy − b) = b2 we got that:(x − 4)(y − 4) = 16

26 / 36 •

M = apq = 4 · 11 · 5 = 220 N = ar = 4 · 71 = 284 Thus, (220, 284)is amicable number by Euler method.

Example

Example Let a = 4 so

b a 4 4 4 4 = = = = = c 2a − σ(a) 2(4) − σ(4) 8 − (1 + 2 + 4) 8 − 7 1

so b = 4 and c = 1. Thus, from (cx − b)(cy − b) = b2 we got that:(x − 4)(y − 4) = 16 x − 4 y − 4 x y p = x − 1 q = y − 1 r = xy − 1 ••• 16 1 20 5 19 4 99 8 2 12 6 11 5 71 4 4 8 8 7 7 63

27 / 36 Example

Example Let a = 4 so

b a 4 4 4 4 = = = = = c 2a − σ(a) 2(4) − σ(4) 8 − (1 + 2 + 4) 8 − 7 1

so b = 4 and c = 1. Thus, from (cx − b)(cy − b) = b2 we got that:(x − 4)(y − 4) = 16 x − 4 y − 4 x y p = x − 1 q = y − 1 r = xy − 1 ••• 16 1 20 5 19 4 99 8 2 12 6 11 5 71 4 4 8 8 7 7 63 •

M = apq = 4 · 11 · 5 = 220 N = ar = 4 · 71 = 284 Thus, (220, 284)is amicable number by Euler method. 28 / 36 • A happy number is mean that the square of the first digit + square of the second digit + ··· + square of last digit = 1 in the end. • 13 is a happy number because 12 + 32 = 10 and 12 + 02 = 1.

Related topics- Happy Amicable Number

• Happy amicable number is an amicable pair (m, n) such that both m and n are happy numbers.

29 / 36 • 13 is a happy number because 12 + 32 = 10 and 12 + 02 = 1.

Related topics- Happy Amicable Number

• Happy amicable number is an amicable pair (m, n) such that both m and n are happy numbers. • A happy number is mean that the square of the first digit + square of the second digit + ··· + square of last digit = 1 in the end.

30 / 36 Related topics- Happy Amicable Number

• Happy amicable number is an amicable pair (m, n) such that both m and n are happy numbers. • A happy number is mean that the square of the first digit + square of the second digit + ··· + square of last digit = 1 in the end. • 13 is a happy number because 12 + 32 = 10 and 12 + 02 = 1.

31 / 36 Happy Amicable Pairs Example

Examples The amicable pair (10572550, 10854650) is happy amicable pair because of that:

10572550 → 12 + 02 + 52 + 72 + 22 + 52 + 52 + 02 = 129 129 → 12 + 22 + 92 = 86 86 → 82 + 62 = 100 100 → 12 + 02 + 02 = 1 10854650 → 12 + 02 + 82 + 52 + 42 + 62 + 52 + 0=167 167 → 12 + 62 + 72 = 86 86 → 82 + 62 = 100 100 → 12 + 02 + 02 = 1

Thus, the amicable pair (10572550, 10854650) is happy amicable pair 32 / 36 • The first three quasi-amicable pairs are (48, 75), (140, 195), and (1050, 1925).

Examples for pair (48, 75) m = 48 and n = 75.

σ(48) = 1 + 2 + 3 + 4 + 6 + 8 + 12 + 16 + 14 + 48 = 124 σ(75) = 1 + 3 + 5 + 15 + 25 + 75 = 124 σ(48) = σ(75) = 48 + 75 + 1 = 124

Thus, the pair (48, 75) is quasi-amicable numbers.

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Quasi-Amicable Pairs

• Quasi-amicable numbers are two positive integers m, n such that the sum of the proper divisors of either number is one more than the value of the other number.

33 / 36 Examples for pair (48, 75) m = 48 and n = 75.

σ(48) = 1 + 2 + 3 + 4 + 6 + 8 + 12 + 16 + 14 + 48 = 124 σ(75) = 1 + 3 + 5 + 15 + 25 + 75 = 124 σ(48) = σ(75) = 48 + 75 + 1 = 124

Thus, the pair (48, 75) is quasi-amicable numbers.

•

Quasi-Amicable Pairs

• Quasi-amicable numbers are two positive integers m, n such that the sum of the proper divisors of either number is one more than the value of the other number. • The first three quasi-amicable pairs are (48, 75), (140, 195), and (1050, 1925).

34 / 36 Quasi-Amicable Pairs

• Quasi-amicable numbers are two positive integers m, n such that the sum of the proper divisors of either number is one more than the value of the other number. • The first three quasi-amicable pairs are (48, 75), (140, 195), and (1050, 1925).

Examples for pair (48, 75) m = 48 and n = 75.

σ(48) = 1 + 2 + 3 + 4 + 6 + 8 + 12 + 16 + 14 + 48 = 124 σ(75) = 1 + 3 + 5 + 15 + 25 + 75 = 124 σ(48) = σ(75) = 48 + 75 + 1 = 124

Thus, the pair (48, 75) is quasi-amicable numbers.

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