The Arm Prime Factors Decomposition

Transcription

1 The Arm Prime Factors Decomosition Arm Boris Nima Abstract We introduce the Arm rime factors decomosition which is the equivalent of the Taylor formula for decomosition of integers on the basis of rime numbers. We make the link between this decomosition and the -adic norm known in the -adic numbers theory. To see how it works, we give examles of these two formulas.

2 Introduction The Arm theory [1] gives the decomosition of functions on each function basis. Also, because the fundamental arithmetic theorem tells us each integers can be decomose in rime factors, I wondered if it was ossible to build the decomosition of integers on the basis of rime integers. As a matter of fact, this is ossible and the answer of this question is the Arm rime factors decomosition. To build an equivalent of the Taylor formula for the rime numbers basis, we need to find a good scalar roduct on this basis. In first lace, I remarked that if a rime number P divise an integer n N + then the difference between it and its integer art would be zero whereas that if does not divise n then the same difference would be ositive. In mathematical words it means : if n then otherwise n n [ ] n [ ] n = 0 > However, we need to build a Kronecker which is one whe is why we take the exonential minus 0.1 multily by x : n δ n = ex is an integer and zero otherwise. This [ ] n x 0.2 which is one when divise n and zero otherwhise. For all that, it still remains the roblem of the multilicity m i of each rime number i because 0.2 tells us only if a rime number is in the rime factors decomosition of an integer n. Thus we have to sum all the ower of if we want to know the multilicity of each factor : δ m n = m i 0.3 With all these ingredients, we build the final Arm rime factors decomosition formula 1.5 which is nothing else that the multilication of 0.3 by the logarithm of all rime numbers. In this case the formula 0.3 gives a -adic valuation in the -adic number theory. Thus we can give an exlicit exression for the well-known -adic norm of rational numbers used in the -adic numbers theory : a = δ m b δ m a 0.4 b for each rime numbers P. 1

3 In the first section, we introduce the Arm rime factors decomosition formula which gives the decomosition of each integer in the rime numbers basis. After describing the scalar roduct, we give in the second section a full examle of rime factors decomosition with the number which is nothing else that Next in the third section, we give the formula of the -adic norm and, in the fourth section, we calculate all -adic norm of the rational which is

4 1 The Arm Prime Factors Decomosition Formula We give the rojection on the rime integers basis of each ositive integer n Theorem 1. n N + the Arm rime factors decomosition is given by lnn = [ ] ex m m x P where [ ] is the integer art and P is the set of all rime numbers. ln 1.5 Proof : The main idea here is that if m n i.e. m divise n then n N + and n m Q + \ N +. In other m words, we have that if m n the is an integer so it is equal to its integer arts. m Because n [ ] n m 0, we can construct the Kronecker m N m δ m n : [ ] ex m m x = 1 if m n [ ] ex m m x = 0 otherwise 1.6 If we consider the usual rime factors decomosition showed in the fundamental theorem of arithmetic : n = i m i 1.8 i=1 where m i is the multilicity of each rime factors i P. In the Arm rime factors decomosition formula, the multilicity m i of each rime factor i is obtained by summing the Kronecker : [ ] ex m m x With 1.9, we can find the Arm rime factors decomosition formula 1.5 : which is the final result. lnn = i = i = P lnn = P m i ln i n ex ex ex m i n m [ n n m m i [ n m 1.7 = m i 1.9 [ n m ] x ln i ] x ln ] x ln

5 Remark 1. We can deduce of the Arm rime factors decomosition formula 1.5 that the scalar roduct of the logarithm of each integer n N + on the basis {ln } P is given by < l, ln > = which the multilicity of each rime factor. [ ] ex m m x 1.11 Corollary 1. Each integer n N + can be decomosed in : n = [ ex n m n ]x m P 1.12 Proof : We just take the exonential of Examle Of Arm Prime Factors decomosition Here we aly the Arm rime factors decomostion formula to find the rime factors of the integer : n = The formula 1.5 gives : lnn = ln2 e n 2 [ n 2 ]x + e n 2 2 [ n 2 2 ]x + e n 2 3 [ n 2 3 ]x ln3 e n 3 [ n 3 ]x + e n 3 2 [ n 3 2 ]x + e n 3 3 [ n 3 3 ]x + e n 3 4 [ n 3 4 ]x ln5 e n 5 [ n 5 ]x + e n 5 2 [ n 5 2 ]x + e n 5 3 [ n 5 3 ]x +e n 5 4 [ n 5 4 ]x + e n 5 5 [ n 5 5 ]x + e n 5 6 [ n 5 6 ]x ln7 e n 7 [ n 7 ]x + e n 7 2 [ n 7 2 ]x + e n 7 3 [ n 7 3 ]x + e n 7 4 [ n 7 4 ]x +e n 7 5 [ n 7 5 ]x + e n 7 6 [ n 7 6 ]x + e n 7 7 [ n 7 7 ]x + e n 7 8 [ n 7 8 ]x

6 which is with the value of n : lnn = ln2 e [ ]x + e [ ]x +e [ ]x ln3 e [ ]x + e [ ]x +e [ ]x + e [ ]x ln5 e [ ]x + e [ ]x + e [ ]x +e [ ]x + e [ ]x + e [ ]x ln7 e [ ]x + e [ ]x + e [ ]x +e [ ]x + e [ ]x +e [ ]x + e [337500]x + e [ ]x When we evaluate, it becomes : lnn = ln e x ln e x ln e 0.8x ln e 0.3x +... So we have the decomosition of lnn in rime factors : lnn = 2 ln2 + 3 ln3 + 5 ln5 + 7 ln Hence If we define the integer function n = = Υ Υi = k k 2.20 k P;k i 5

7 3 Link with the -adic numbers The formula 1.9 of an integer n gives the multilicity of an rime factor. This multilicity is called the -adic valuation in the -adic numbers theory. We give here its exlicit exression Proosition 1. The -adic valuation of a rime factor i of an integer n is given by [ ] v n = ex m m x Proof : See 1.9. Now, with the -adic valuation, we can define the corresonding -adic norm Proosition 2. The -adic norm of a rational number a b Q, where a and b are corime, is given by a [ [ = ex b m b ]x m ex a m a ]x m 3.22 b or its logarithm is : a ln = b Proof : The -adic norm of an integer a : [ ] [ ] b b a a ex m m x ex m m x a = i ln 3.23 i mi 3.24 as in 1.8, where the rime factor i P and m i is the multilicity of each i, is defined as a i = m i i 3.25 However we know from 1.9 that the multilicity of each rime factor is given by i [ ] a a ex m m x = m i 3.26 Hence the -adic norm of n 3.25 is given by a i = [ ex a m a ]x m i 3.27 And so the -adic norm of a rational a b Q is given by a [ b = ex b m b ]x m i i which gives the formula ex [ a m a ]x m

8 4 Examle of -adic norm As examle, we consider the rational : a = 4.29 b With the formula 3.22, we can first calculate the 2-adic norm [ ] [ ] ln = ex 2 2 m 2 m x ex 2 m 2 m x ln 2 [ ] [ ] = ex 2 x ex 2 2 x ln 2 2 ln = ln or in the exonential form : = Now we calculate the 3-adic norm ln = 3 = + ln 3 or in the exonential form : Now we calculate the 5-adic norm ln = 5 = ex ex ex [ 3 m 3 m [ ] 3 x ex 3 [ ] 9 x ex 9 ] x ex [ ] 3 m 3 m x ln 3 [ ] 3 x ln 3 3 [ ] 9 x ln 3 9 = 2 ln ln 5 or in the exonential form : ex ex = ex [ 5 m 5 m [ ] 5 x ex 5 [ ] 25 x ex 25 ] x ex 4.33 [ ] 5 m 5 m x ln 5 [ ] 5 x ln 5 5 [ ] 25 x ln 5 25 = 2 ln =

9 Now we calculate the 7-adic norm ln = 7 ln 7 = or in the exonential form : ex Now we calculate the 11-adic norm ln = 11 ln 11 ex 7 [ 7 m 7 m [ ] x 7 ] x ex ex 7 [ 7 m 7 m [ ] x 7 ] x ln 7 ln 7 = ln = or in the exonential form : ex = ex 11 [ 11 m 11 m [ ] x 11 ] x ex ex [ ] 11 m 11 m x ] x ln 11 [ 11 ln 11 = ln = And we have that = {2, 3, 5, 7, 11}. So we have the rime factors decomosition of a b =

10 Discussion Maybe the Arm rime factors formula 1.5 is too simle but it gives a ractical way to calculate the decomosition of every logarithms of integers on the basis of the logarithm of rime numbers. However, the Arm rime factors decomosition formula 1.5 is not efficient when it is rogrammed on comuters because it does the same work as the traditional algorithm it checks if the division is an integer or not. In addition, the traditional algorithm which do the rime factorization is faster than this one. The only utility of my formula is that it gives a ractical formula in the theory. Furthermore in the formula 1.5 there is summation over ositive integers and we have to sto it until a big value if we want the algorithm to be finish. Besides there is an other summation over rime numbers and the algorithm needs to calculate all the rime numbers so it takes a lot of time for calculating. After all, I ve decided to write this article even if the algorithm is not efficient because the formulas are right and give the results. 9

11 Références [1] Arm B. N., The Arm Theory 10