The Anti-Divisor

The anti-divisor, or unbiased non-divisor, is a concept related very closely to the concept of prime numbers, and the concept of a divisor.

Every integer is either prime, or has two or more prime factors, for example 61 is prime, but 63 can be written as 3*3*7.

Every integer is then said to be the product of some factors. A divisor is a combination of these factors, for example 63 has the divisors 1,3,7,9,21 and 63, as 63 divided by any of these numbers yields an integer.

It logically follows from this that any number that is not a divisor of an integer is a non-divisor. So, we say that the non-divisors of 63 are all the integers less than or equal to 63 except for 1,3,7,9,21 and 63.

The definition of an anti-divisor follows on from this - an anti-divisor is a non-divisor such that doesn't divide the number in the most unbiased way possible. For example, we say 42 is an anti-divisor of 63, as 42 surrounds 63 with a gap of 21 on either side. 41 on the other hand, has gaps of 22 and 18, so the gap of 22 is larger than the 18. 41 is called a biased non-divisor of 63.

There are two distinct mathematical definitions of anti-divisors, one for even anti-divisors, and one for odd anti-divisors. The two definitions are very similar, however we need two definitions because a even anti-divisor candidate defines exactly one number, and an odd anti-divisor candidate defines exactly two numbers.

Both definitions do share a common feature. An anti-divisor of n must lie in the region [2,n-1]. 1 is never an anti-divisor, its exclusion is determined by the phrase 'non-divisor', and the fact that no integers are contained between two successive integers with 1 as a divisor. Also note that there are 'larger-than'n' anti-divisors, namely 2n-1, 2n and 2n+1, but these are considered to be trivial.

For k even:

k is an even anti-divisor of n when nmodk=k/2.

Or to put it another way, k is an even anti-divisor of n when k(x+1/2)=n, for some x greater than or equal to 1.

For example 10 is an even anti-divisor for 15, 25, 35, 45, 55, etc,.. and no other numbers.

This is because 10(x+1/2)=10x+5, and so the numbers are generated over x>1.

Another example, 8 is an even anti-divisor for 12,20,28,etc..., and no other numbers.

For k odd:

k(x+1/2) falls between 2 integers, and we claim that both of these integers are anti-divisors for n.

Hence the basic definition is the same for odd anti-divisors, except that k(x+1/2) itself is never an integer, so we define the odd anti-divisors as the two numbers surrounding a k(x+1/2).

So k is an odd anti-divisor of n when we have either k(x+1/2)-1/2 or k(x+1/2)+1/2 for some x greater than or equal to 1.

In mod terminology, k is an odd anti-divisor of n when nmodk=(k-1)/2 or nmodk=(k+1)/2.

So, 7 is an odd anti-divisor for 10,11,17,18,23,24, etc.., and no other numbers.

This is because we have the two generating functions 7x+3 and 7x+4.

Again, 11 is an odd anti-divisor for 11x+5, 11x+6, i.e. 16,17,27,28, etc..., and no other numbers.

Putting it together

Now we have defined an anti-divisor, let's consider an integer n, e.g.10.

We then find all the anti-divisors of 10, in this case 3,4 and 7.

Spiel

The pattern of anti-divisors is as random and incomprehensible as that with prime numbers.

The anti-divisors of the first few integers are:

nAnti-divisors
2none
32
43
52.3
64
72.3.5
83.5
92.6
103.4.7
112.3.7
125.8
132.3.5.9
143.4.9
152.6.10
163.11
172.3.5.7.11
184.5.7.12
192.3.13
203.8.13
212.6.14
223.4.5.9.15
232.3.5.9.15
247.16
252.3.7.10.17
263.4.17
272.5.6.11.18
283.5.8.11.19
292.3.19
304.12.20
312.3.7.9.21
323.5.7.9.13.21
332.5.6.13.22
343.4.23
352.3.10.14.23
368.24
372.3.5.15.25
383.4.5.7.11.15.25
392.6.7.11.26
403.9.16.27
412.3.9.27
424.5.12.17.28
432.3.5.17.29
443.8.29
452.6.7.10.13.18.30
463.4.7.13.31
472.3.5.19.31
485.19.32
492.3.9.11.14.33
503.4.9.11.20.33
512.6.34
523.5.7.8.15.21.35
532.3.5.7.15.21.35
544.12.36
552.3.10.22.37
563.16.37
572.5.6.23.38
583.4.5.9.13.23.39
592.3.7.9.13.17.39
607.8.11.17.24.40
612.3.11.41
623.4.5.25.41
632.5.6.14.18.25.42
643.43
652.3.10.26.43
664.7.12.19.44
672.3.5.7.9.15.19.27.45
683.5.8.9.15.27.45
692.6.46
703.4.20.28.47
712.3.11.13.47
725.11.13.16.29.48
732.3.5.7.21.29.49
743.4.7.21.49
752.6.10.30.50
763.8.9.17.51
772.3.5.9.14.17.22.31.51
784.5.12.31.52
792.3.53
803.7.23.32.53
812.6.7.18.23.54
823.4.5.11.15.33.55
832.3.5.11.15.33.55
848.13.24.56
852.3.9.10.13.19.34.57
863.4.9.19.57
872.5.6.7.25.35.58
883.5.7.16.25.35.59
892.3.59
904.12.20.36.60
912.3.14.26.61
923.5.8.37.61
932.5.6.11.17.37.62
943.4.7.9.11.17.21.27.63
952.3.7.9.10.21.27.38.63
9664
972.3.5.13.15.39.65
983.4.5.13.15.28.39.65
992.6.18.22.66
1003.8.40.67

[More]

From here we can make a few initial observations. Every integer has a largest anti-divisor, and this is at approximately at 2/3rds of n. This can be used to prove that every number has a unique set of anti-divisors. [Proof]

Anti-divisors can be used to prove that there are an infinite number of primes.[Proof]

Anti-primes (integers with only one anti-divisor) are rare, and include 3,4,6,96 and 393216.[More info]

The most noticeable result from anti-divisors is when we consider anti-perfect numbers. These are integers such that the sum of its anti-divisors equals the original integers. These start of with 5,8,41,56,946. [Larger list]

There is a simple formula for finding anti-divisors [Formula] and this formula can be used to obtain the theorem that an integer (2k+1) is prime iff k and k+1 do not share any anti-divisors.

It is also possible to derive a very simple method for generating primes numbers [Primes].

Another exciting feature of anti-divisors is that each integer displays a unique decomposition pattern, and more, each integer ultimately decomposes into the empty set. There is more information, and an applet showing the varying decomposition stages. [Decomposition]

And more: k-biased divisors, anti-divisor rarities, semi-anti-perfect numbers, MINOR and MAJOR numbers, etc. [More]

And even more: Anti-divisor sums, etc...[Even More]

[Story of the Anti-Divisor]

[Other maths]

People who have contributed towards this site: [Contributors]

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