Talk:Amenable number

Definition

The definition in the article does not agree with that of Clifford A. Pickover (2011). A Passion for Mathematics: Numbers, Puzzles, Madness, Religion, and the Quest for Reality. John Wiley & Sons. p. 88. ISBN 1118046072. where the sum and the product have to have the same number of terms but that number does not have to be the target. So 4 = 2+2 = 2×2 is explicitly stated to be amenable. Deltahedron (talk) 09:11, 8 December 2012 (UTC)

Pickover may have gotten his definition from an earlier version of this page or an earlier version of the MathWorld page. The original definition definitely required the number of terms to be the target. Without that requirement every natural number is amenable, just use {1,1,-1,-1,n} for the natural number n. This page was rewritten when the MathWorld page was revised, but some parts of our page still reflect the older version's definition. I will attempt to straighten this out. Bill Cherowitzo (talk) 04:51, 6 August 2014 (UTC)

Question

Could this concept of "amenable number" have anything to do with the algebraic concept of "amenable group" ??

In mathematics, an amenable group is a locally compact topological group G carrying a kind of averaging operation on bounded functions that is invariant under translation by group elements. The original definition, in terms of a finitely additive invariant measure (or mean) on subsets of G, was introduced by John von Neumann in 1929 under the German name "messbar" ("measurable" in English) in response to the Banach–Tarski paradox. In 1949 Mahlon M. Day introduced the English translation "amenable", apparently as a pun.

Anyway, are these two concepts linked together ?? Or will there be nothing? — Preceding unsigned comment added by 179.178.220.101 (talk) 12:11, 26 January 2017 (UTC)

Amenable Numbers Categorization

The following is the Categorization of amenable numbers: 1. Numbers of form 4n are amenable if either they are divisible by 8 or have a factor of form 4n + 3. So, other than 4, there are many other exceptions like 20, 52, etc. The set in case they are divisible by 8 is formed by a factor of form 4n and another of form 4n + 2. If they have a factor of form 4n + 3 and are not divisible by 8, then the set uses three of its factors- two same or different factors of form 4n + 2 and a third odd factor of form 4n + 3. 2. Numbers of form 4n + 1 are always amenable with its every possible integer factorization. They are of form 4n + 1 multiplied any number of times * 4n + 3 even number of times. Prime numbers of form 4n + 1 have the unique trivial amenable set: {-1 (2n times), 1 (2n times), number itself} 3. No number of form 4n + 2 is amenable because there is only one even number and there are odd number (4n + 1) of other terms which are all odd and so, the sum will be odd. 4. No number of form 4n + 3 is amenable because of their prime factorization properties. They are of form 4n + 1 multiplied any number of times * 4n + 3 odd number of times. Note: Here, we are talking about modulus of 4 of a number and so, all numbers of same modulus and variable are not necessarily equal. Usage of same modulus and variable is just for simplicity. Usermaths (talk) 07:28, 8 November 2024 (UTC)

Just what the world needs: more AI hallucinations posted on Wikipedia talk-pages. 100.36.106.199 (talk) 10:14, 8 November 2024 (UTC)

This is not AI. I am an human being. Why are you joking 100.36.106.199? Usermaths (talk) 12:18, 8 November 2024 (UTC)

Why do you think it is false information? This is correct information using pairs of -1 and 1, 1s and integral factors of the amenable number. I have checked it thoroughly and repeatedly. Why do you think I am adding wrong info in talk page? Usermaths (talk) 12:22, 8 November 2024 (UTC)

I have proofs and here they are: Usermaths (talk) 13:20, 8 November 2024 (UTC)

Every number of form 4n + 1 is an amenable number because of the set:

{1 (2n times), -1 (2n times), number itself}

No number of form 4n + 2 is an amenable number because of they are = 2 * (2n + 1). So, in case of product only one term is even and all others are odd. So, there are 4n + 1 odd numbers in the set and since 4n + 1 is an odd number, then the sum of the numbers is an odd number which is definitely not of form 4n + 2.

Every number divisible by 8 are also amenable numbers. So, they can be expressed as product of two numbers, one of form 4n and other of form 4n + 2. Their sum is also of form 4n + 2. The number of remaining numbers is 8n - (4n + 2) which is of form 4n + 2. No of 1s to be added is 8n - (4n + 2) which is again 4n + 2. So, add (4n + 2) 1s. The remaining number of places is 8n - 4n = 4n and their goes pairs of 1 and -1 which will yield positive product. Note: all n's are not necessarily equal.

Example:

8 = 2 * 4; 2 + 4 = 6

8 - 6 = 2; 8 - 2 - 2 = 4

So, the set contains:

{-1, 1, -1, 1, 1, 1, 2, 4}

24 = 4 * 6 = 2 * 12

4 + 6 = 10 and 2 + 12 = 14

24 - 10 = 14 and 24 - 14 = 10; 24 - 2 - 14 = 8 and 24 - 2 - 10 = 12

So the two sets are:

{-1, 1, -1, 1, -1, 1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 6} and

{-1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 12}

Any composite number of form 4n + 1 can be written as a product of either (4n + 1) * (4n + 1) * (4n + 1) or (4n + 1) * (4n + 3) * (4n + 3). Sum of factors = 4n + 1 + 4n + 3 + 4n + 3 = 4n + 1 + 4n + 1 + 4n + 1 = 4n + 3. (4n + 1) - (4n + 3) = 4n + 2. Add 4n + 2 1s. 4n + 2 + 3 = 4n + 1 places used up. So, (4n + 1) - (4n + 1) = 4n places left. Add pairs of -1 and 1 there. Prime Numbers of form 4n + 1 fail to have any non-trivial representation. Note: all n's are not necessarily equal.

Examples:

9 = 1 * 3 * 3

1 + 3 + 3 = 7; 9 - 7 = 2

2 + 3 = 5; 9 - 5 = 4

So, the set is:

{-1, 1, -1, 1, 1, 1, 1, 3, 3}

45 = 1 * 5 * 9 = 3 * 3 * 5

1 + 5 + 9 = 15

3 + 3 + 5 = 11

45 - 15 = 30; 45 - 11 = 34

30 + 3 = 33; 34 + 3 = 37

45 - 33 = 12; 45 - 37 = 8

So, the sets are:

{-1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 9} and

{-1, 1, -1, 1, -1, 1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 5}

Numbers of form 4n which are not divisible by 8 and has a factor of form 4n + 3 are always amenable. They can be expressed as (4n + 2) * (4n + 2) * (4n + 3). Sum of factors = 4n + 2 + 4n + 2 + 4n + 3 = 4n + 3. 4n - (4n + 3) = 4n - 3 which is of form 4n + 1. Add 4n + 1 1s. These numbers take 3 + 4n + 1 = 4n places. In the remaining 4n - 4n = 4n places, add pairs of -1 and 1. Note: all n's are not necessarily equal.

Examples

12 = 2 * 2 * 3

2 + 2 + 3 = 7; 12 - 7 = 5

5 + 3 = 8; 12 - 8 = 4

So, the set is:

{-1, 1, -1, 1, 1, 1, 1, 1, 1, 2, 2, 3}

36 = 2 * 6 * 3

2 + 6 + 3 = 11; 36 - 11 = 25

25 + 3 = 28; 36 - 28 = 8

So, the set is:

{-1, 1, -1, 1, -1, 1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 6}

Numbers of form 4n + 3 are not amenable. As they can be written as:

(4n + 1) repeated any number of times * (4n + 3) repeated odd number of times

Note: all 4n + 1 and all 4n + 3 are not necessarily equal.

Numbers of form (4n + 1) repeated 4n times so, sum is 4n. For any 4n + x times repetition of 4n + 1, the sum is 4n + x. For 4n + 3 repeated 4n + 1 number of times gives 4n + 3 and 4n + 3 repeated 4n + 3 number of times gives 4n + 1. Because of the Inversion of sum and repetition modulo 4 in 4n + 3 odd number of times, you have to place 4n + x 1s from remaining 4n + x - 2 places where we have to add 2n + 1 pairs of -1 and 1 in remaining 4n + 2 places and the product will be negative.

But, in case of numbers of form 4n + 1, they can be represented as (4n + 1) repeated any number of times * (4n + 3) repeated even number of times

Here, 4n + 3 repeated 4n + 2 number of times gives 4n + 2 and 4n + 3 repeated 4n number of times gives 4n. Here, modulus is preserved so, we have to add 4n + x 1s in 4n + x places resulting in 2n pairs of -1 and 1 placed in remaining 4n places and the product will remain positive. So, represent numbers of form 4n + 1 as product of odd numbers of any type (4n + 1 or 4n + 3) using any number of odd numbers, you will have one corresponding amenable set (maximum number possible is the number of prime factors of the odd number containing all its prime factors).

Examples

21 = 21 = 3 * 7

The one odd number will give the trivial set:

{-1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 21}

3 + 7 = 10; 21 - 10 = 11

11 + 2 = 13; 21 - 13 = 8

So, the set is:

{-1, 1, -1, 1, -1, 1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 7}

Note : all n's are not necessarily equal, all numbers of form 4n + 1 or 4n + 3 in the prime factorizations are not necessarily equal. So, every number of form 4n + 1 is amenable and no number of form 4n + 3 are amenable.

Numbers of form 4n not divisible by 8 and do not have a factor of form 4n + 3 are never amenable. They can be expressed as:

4n * (4n + 1) repeated any number of times

And (4n + 2) * (4n + 2) * (4n + 1) repeated any number of times

The first representation contains only one even number which makes the sum odd in the set and so, this is not an amenable set. The second representation uses 4n + 1 repeated 4n + x times and so has sum 4n + x. The two 4n + 2 adds up to 4n and the sum remains 4n + x but the repetition becomes 4n + x + 2 and because of this you have to add 4n + x - 2 1s in 4n + x places and remaining 4n + 2 places have 2n + 1 pairs of -1 and 1 resulting in negative product. So, they are never amenable.

So, for summary:

1. Numbers of form 4n are amenable if they are divisible by 8 and the set uses it's even factors 4n and 4n + 2. If the number is not divisible by 8 and has a factor of form 4n + 3, then the set uses it's even factors 4n + 2 and 4n + 2 and a factor of form 4n + 3. If neither property holds (divisibility by 8 or having a factor of form 4n + 3), they are not amenable.

2. Numbers of form 4n + 1 are always amenable as the set can use any number of its factors to make itself.

3. Numbers of form 4n + 2 are never amenable because of resulting odd sum in the set.

4. Numbers of form 4n + 3 are never amenable because of their prime factorization and modular inversion resulting in negative product. Usermaths (talk) 13:31, 8 November 2024 (UTC)

Primes are also amenable but they are trivially amenable. The trivial set is:

{-1 (2n times), 1 (2n times), number itself} Usermaths (talk) 13:33, 8 November 2024 (UTC)

The trivial set is for Primes of form 4n + 1. Usermaths (talk) 13:36, 8 November 2024 (UTC)

What is currently in the article is correct and is supported by sources. What you have written is not correct and is in direct violation of WP:OR, which is one of the foundational policies on which Wikipedia is based. Have you read WP:OR? 100.36.106.199 (talk) 17:39, 8 November 2024 (UTC)