A RECREATION AROUND THE GOLDBACH

1 downloads 1 Views 299KB Size Report
Indeed, the Fermat last problem was solved by A. Wiles ( (see [1]) and a logic (non recursive) proof of the Fermat last problem was given by Ikorong Nemron ...

ASIAN JOURNAL OF MATHEMATICS AND APPLICATIONS Volume 2013, Article ID ama0004, 6 pages ISSN 2307-7743 http://scienceasia.asia

A RECREATION AROUND THE GOLDBACH PROBLEM AND THE FERMAT LAST PROBLEM IKORONG ANOUK GILBERT NEMRON

Abstract. The Goldbach problem ( see [2] or [3] or [4] or [5] or [6] or [7] or [8] or [9] or [10] or [11] or [12] or [13] or [14] or [15] or [20] or [21] or [22] or [23] ) states that every even integer e > 4 is of the form e = p + q, where (p, q) is a couple of prime(s). Indeed, a logic (non recursive) proof of the Goldbach problem was given by Ikorong Nemron (see [16]). The Fermat last problem (see [13] or [15] or [19]) stipulates that when n is an integer > 3, the equation xn + y n = z n has not solution in integers > 1. Indeed, the Fermat last problem was solved by A. Wiles ( (see [1]) and a logic (non recursive) proof of the Fermat last problem was given by Ikorong Nemron ( see [17] or [18]). That being so, in this paper, we present two simple analytic reformulations of the Goldbach problem and the Fermat last problem via prime numbers; and using these two simple analytic reformulations, we show that the Goldbach problem and the Fermat last problem can be restated in way that clearly resembles each other. More precisely, using these two simple analytic reformulations of the Goldbach problem and the Fermat last problem via prime numbers, we present a curious strong analytic resemblance between the Goldbach problem and the Fermat last problem; based on this resemblance, we explain why it is natural and not surprising that the Goldbach problem and the Fermat last problem can be solved simultaneously by analytic argument on prime numbers.

Preliminary. This paper is an original investigation concerning the analytic attack of the Fermat last problem and the Goldbach problem, since this analytic attack is very simple to understand. That being so, this paper is divided into two simple Sections. In Section.1, we introduce definitions that are not standard and we present some elementary properties. In Section.2, using definitions of Section.1, we show two simple Theorems which are clearly resembling and which are simple analytic reformulations of the Goldbach problem and the Fermat last problem using prime numbers. Using these two simple analytic reformulations, it becomes immediate to see that the Goldbach problem and the Fermat last problem can be restated in way that clearly resembles each other, so that it is natural and not surprising that the Goldbach problem and the Fermat last problem can be solved simultaneously by analytic argument on prime numbers. We start. 1. Non-standard definitions and simple properties. In this section, we introduce definitions that are not standard. These definitions are necessary for the final two analytic reformulations of the Goldbach problem and the Fermat last problem using prime numbers. 2010 Mathematics Subject Classification. 11P32; 11D41. c

2013 Science Asia

1

2

NEMRON

We say that e is goldbach, if e is an even integer > 4 and is of the form e = p + q, where (p, q) is a couple of prime(s). The Goldbach problem (see Abstract) states that every even integer e > 4 is goldbach [Example.0. 100 is goldbach, because 100 is an even integer > 4 and is of the form 100 = 53 + 47, where 53 and 47 are prime]. We say that e is goldbachian, if e is an even integer > 4, and if every even integer v with 4 6 v 6 e is goldbach [Example.1. 100 is golbachian. Indeed, 100 is an even integer > 4, and it is easy to check that every even integer v of the form 4 6 v 6 100 is goldbach. Note that goldbachian implies goldbach; so there is not confusion between goldbachian and goldbach]. Now, for every integer n > 2,

we define G(n) and gn as follows: G(n) = {g; 1 < g 6 2n, and g is goldbachian}, and gn = max g. From the definition of G(n) and gn , we immediately deduce: g∈G(n)

G(n+1) = {g; 1 < g 6 2n+2, and g is goldbachian}, and gn+1 =

max

g. In Sec-

g∈G(n+1)

tion.2, gn+1 will help us to give an analytic simple reformulation of the Goldbach problem via prime numbers. From the definition of gn and gn+1 , it is immediate to see. Proposition 1.0. Let n be an integer > 2. We have the following two simple properties. (1.0.0.) gn and gn+1 are even; gn 6 2n; and gn 6 gn+1 6 2n + 2. (1.0.1.) gn+1 = 2n + 2 if and only if 2n + 2 is goldbachian. Proof. Properties (1.0.0.) and (1.0.1.) are immediate [it suffices to use the definition of gn and gn+1 ]. Proposition 1.1. Let n be an integer > 2. Then we have the following three simple properties. (1.1.0.) If 2n + 2 is not goldbachian, then n > 2 and for every integer u > n, 2u + 2 is not goldbachian. (1.1.1.) If gn+1 < 2n + 2 [i.e. if 2n + 2 is not goldbachian], then n > 2 and for every integer u > n, we have gu+1 = gn+1 = gn . (1.1.2.) If gn+1 = Z, where Z < 2n + 2, then limy→+∞ gn+1+y = Z. Proof. Property (1.1.0.) is immediate [it suffices to apply the definition of goldbachian ]; property (1.1.1.) is an immediate reformulation of Property (1.1.0.), and property (1.1.2) is an obvious consequence of property (1.1.1). That being so, we say that e is wiles, if e is an integer > 3 and if the equation xe + y e = z e has not solution in integers > 1. The Fermat last problem states that every integer e > 3 is wiles [Example.2. it is known that 3, 4, 5 and 6 are all wiles]. We say that e is wilian’s, if e is an integer > 3, and if every integer v with 3 6 v 6 e is wiles [Example.3. 6 is wilian’s. Indeed, 6 is an integer > 3, and using Example.2., we see that every integer v of the form 3 6 v 6 6 is wiles. Note that wilian’s implies wiles; so there is not confusion between wilian’s and wiles]. Now, for every integer n > 3, we define W(n) and wn as follows: W(n) = {w; 2 < w 6 n, and w is wilian 0 s}, and

wn = 2 max w. From the definition of W(n) and wn , we immediately deduce: w∈W(n)

W(n + 1) = {w; 2 < w 6 n + 1, and w is wilian 0 s}, and wn+1 = 2

max

w. In

w∈W(n+1)

Section.2, wn+1 will help us to give an analytic reformulation of the Fermat last problem via prime numbers. From the definition of wn and wn+1 , it is immediate to see. Proposition 1.2. Let n be an integer > 3. We have the following two simple properties. (1.2.0.) wn and wn+1 are even; wn 6 2n; and wn 6 wn+1 6 2n + 2. (1.2.1.) wn+1 = 2n + 2 if and only if n + 1 is wilian’s.

A RECREATION

3

Proof. Properties (1.2.0.) and (1.2.1.) are immediate [it suffices to use the definition of wn and wn+1 ].

Observe that Proposition 1.2 resembles to Proposition 1.0. Proposition 1.3. Let n be an integer > 3. Then we have the following three simple properties. (1.3.0.) If n + 1 is not wilian’s, then n > 3 and for every integer u > n, u + 1 is not wilian’s. (1.3.1.) If wn+1 < 2n + 2 [i.e. if n + 1 is not wilian’s], then n > 3 and for every integer u > n, we have wu+1 = wn+1 = wn . (1.3.2.) If wn+1 = Z, where Z < 2n + 2, then limy→+∞ wn+1+y = Z. Proof. Property (1.3.0.) is immediate [it suffices to apply the definition of wilian’s and to observe that, if n = 3, then n + 1 is wilian’s]; property (1.3.1.) is an immediate reformulation of Property (1.3.0.), and property (1.3.2) is an obvious consequence of property (1.3.1). Observe that Proposition 1.3 resembles to Proposition 1.1. Prime numbers are well known. Now for every integer n > 2, we define P(n) and pn as follows: P(n) = {p; p is prime and 1 < p < 2n}, and pn = max p. p∈P(n)

Using the definition of pn , it is known: Theorem 1.4 (The Postulate of Bertrand or Erdos Theorem). Let n be an integer > 1, then there exists a prime between n and 2n.  Corollary 1.5. For every integer n > 2, pn > n. Proof. Use definition of pn and Theorem 1.4.  2. An analytic resemblance between the Goldbach problem and the Fermat last problem. In this section, we show that the Goldbach problem and the Fermat last problem can be restated in ways that resemble each other. More precisely, we show two simple Theorems which are clearly resembling and which are simple analytic reformulations of the Goldbach problem and the Fermat last problem using prime numbers. That being so, using the definition of goldbachian, then the following first Theorem is an analytic simple reformulation of the Goldbach problem via prime numbers. Theorem 2.0 (An analytic reformulation of the Goldbach problem via prime numbers). The following are equivalent. (1) The Goldbach problem is true [i.e. every even integer e > 4 is of the form e = p + q, where (p, q) is a couple of prime(s)]. (2) For every integer n > 2, 2n + 2 is goldbachian. (3) For every integer n > 2, gn+1 = 2n + 2. (4) For every integer n > 2, gn > pn . (5) For every integer n > 2, gn > n. (6) limn→+∞ gn+1 = +∞. To prove easily Theorem 2.0, let us remark. Remark.0 The following are equivalent. (i) The Goldbach problem is true. (ii) limn→+∞ gn+1 = +∞. Proof. (i) ⇒ (ii)]. Immediate [ it suffices to use the definition of the Goldbach problem]. (ii) ⇒ (i)]. Otherwise, let M be a finite integer such that 2M + 2 is not goldbachian [ such a M clearly exists, since the Goldbach problem is false]; since 2M + 2 is not goldbachian, clearly

4

NEMRON

gM +1 6= 2M + 2 Observing [ by using the definition of gM +1 ] that gM +1 6 2M + 2 then, using (R.0.0) and (R.0.1), we immediately deduce that gM +1 < 2M + 2

(R.0 .0 ).

(R.0 .1 ),

(R.0 .2 ).

Inequality (R.0.2) clearly says that gM +1 = Z, where Z < 2M + 2 (R.0 .3 ); it is immediate that Z is a finite integer, since M is a finite integer. Now using (R.0.3) and property (1.1.2) of Proposition 1.1, then we immediately deduce that limn→+∞ gM +1+n = Z, where Z is a finite integer; the previous immediately implies that limn→+∞ gn+1 = Z, where Z is a finite integer, and this is absurd, since limn→+∞ gn+1 = +∞ [ by the hypothese]. So, assuming that the Goldbach problem is false gives rise to a serious contradiction; so the Goldbach problem is true.

Proof of Theorem 2.0. (1) ⇒ (2)]. Immediate [it suffices to use the definition of the Goldbach problem and the definition of goldbachian]; (2) ⇒ (3)] Immediate [it suffices to use the definition of goldbachian and the definition of gn+1 ]; (3) ⇒ (4)] Immediate [it suffices to observe (by using the definition of pn ) that pn 6 2n − 1]; (4) ⇒ (5)] Immediate [it suffices to use Corollary 1.5]; (5) ⇒ (6)] Indeed, observing [by the hypothese] that for every integer n > 2 we have gn+1 > n, clearly limn→+∞ gn+1 = +∞; (6) ⇒ (1)] Observing [by the hypothese] that limn→+∞ gn+1 = +∞, then, using Remark.0, we immediately deduce that the Goldbach problem is true.  Now using the definition of wilian’s, then the following Theorem is the corresponding analytic reformulation of the Fermat last problem. Theorem 2.1. ( An analytic reformulation of the Fermat last problem). The following are equivalent. (1) The Fermat last problem is true [i.e. for every integer n > 3, the equation xn + y n = z n has not solution in integers > 1]. (2) For every integer n > 3, n + 1 is wilian’s. (3) For every integer n > 3, wn+1 = 2n + 2. (4) For every integer n > 3, wn+1 > pn . (5) For every integer n > 3, wn+1 > n. (6) limn→+∞ wn+1 = +∞. To prove easily Theorem 2.1, let us remark. Remark.1 The following are equivalent. (i) The Fermat last problem is true. (ii) limn→+∞ wn+1 = +∞. Proof. (i) ⇒ (ii)]. Immediate [it suffices to use the definition of the Fermat last problem]. (ii) ⇒ (i)]. Otherwise, let M be a finite integer such that M + 1 is not wilian’s [ such a M exists, since the Fermat last problem is false]; since M + 1 is not wilian’s, clearly wM +1 6= 2M + 2 Observing [ by using the definition of wM +1 ] that

(R.1 .0 ).

wM +1 6 2M + 2 then, using (R.1.0) and (R.1.1), we immediately deduce that

(R.1 .1 ),

wM +1 < 2M + 2

(R.1 .2 ).

Inequality (R.1.2) clearly says that wM +1 = Z, where Z < 2M + 2 (R.1 .3 ); it is immediate that Z is a finite integer, since M is a finite integer. Now using (R.1.3) and property (1.3.2) of Proposition 1.3, then we immediately deduce that limn→+∞ wM +1+n = Z, where Z is a finite

A RECREATION

5

integer; the previous immediately implies that limn→+∞ wn+1 = Z, where Z is a finite integer, and this is absurd, since limn→+∞ wn+1 = +∞ [ by the hypothese]. So, assuming that the Fermat last problemis false gives rise to a serious contradiction; so the Fermat last problemis true.

Observe that Remark.1 and Remark.0 are resembling. Proof of Theorem 2.1. (1) ⇒ (2)]. Immediate [it suffices to use the definition of the Fermat last problemand the definition of wilian’s]; (2) ⇒ (3)] Immediate [it suffices to use the definition of wilian’s and the definition of wn+1 ]; (3) ⇒ (4)] Immediate [it suffices to observe (by using the definition of pn ) that pn 6 2n − 1]; (4) ⇒ (5)] Immediate [it suffices to use Corollary 1.5]; (5) ⇒ (6)] Indeed, observing [by the hypothese] that for every integer n > 3 we have wn+1 > n, clearly limn→+∞ wn+1 = +∞; (6) ⇒ (1)] Observing [by the hypothese] that limn→+∞ wn+1 = +∞, then, using Remark.1, we immediately deduce that the Fermat last problem is true.  Visibly, Theorem 2.1 and Theorem 2.0 are strongly resembling; so resembling that they seem identical. Using Theorem 2.1 and Theorem 2.0 , it becomes immediate to see that the Goldbach problem and the Fermat last problem can be restated in way that clearly resembles each other, so that it is natural and not surprising that the Goldbach problem and the Fermat last problem can be solved simultaneously by analytic argument on prime numbers. References. [1] A. Wiles. Modular Elliptic Curbes And Fermat’s Last Theorem. Annals Of Mathematics 141. 443 − 551, (1995). [2] A. Schinzel, Sur une consequence de l’hypoth` ese de Goldbach. Bulgar. Akad. Nauk. Izv. Mat. Inst.4, (1959). 35 − 38. [3] Bruce Schechter. MY BRAIN IS OPEN (The mathematical journey of Paul Erdos) (1998). 10 − 155. [4] Dickson. THEORY OF NUMBERS (History of Numbers. Divisibity and primality) Vol 1. Chelsea Publishing Company. New York , N.Y (1952). Preface.III to Preface.XII. [5] Dickson. THEORY OF NUMBERS (History of Numbers. Divisibity and primality) Vol 1. Chelsea Publishing Company. New York , N.Y (1952) [6] G.H Hardy, E.M Wright. An introduction to the theory of numbers. Fith Edition. Clarendon Press. Oxford. [7] Ikorong Anouk Gilbert Nemron An alternative reformulation of the Goldbach problem and the twin primes conjecture. Mathematicae Notae. Vol XLIII (2005). 101 − 107. [8] Ikorong Anouk Gilbert Nemron. Around The Twin Primes Conjecture And The Goldbach Conjecture I. Tomul LIII, Analele Stiintifice Ale Universitatii ”Sectiunea Matematica”. (2007). 23 − 34. [9] Ikorong Anouk Gilbert Nemron. An original symposium over the Goldbach conjecture, The Fermat primes, The Fermat composite numbers conjecture, and the Mersenne primes conjecture .Mathematicae Notae. Vol XLV (2008). 31 − 39. [10] Ikorong Anouk Gilbert Nemron. An Original Reformulation Of The Goldbach Conjecture. Journal of Discrete Mathematical Sciences And Cryptography; Taru Plublications; Vol.11;(2008). Number 4, 465 − 469. [11] Ikorong Anouk Gilbert Nemron. An original abstract over the twin primes, the Goldbach Conjecture, the Friendly numbers, the perfect numbers, the Mersenne composite numbers, and the Sophie Germain primes. Journal of Discrete Mathematical Sciences And Cryptography; Taru Plublications; Vol.11; Number.6, (2008). 715 − 726. [12] Ikorong Anouk Gilbert Nemron. Playing with the twin primes conjecture and the Goldbach conjecture. Alabama Journal of Maths; Spring/Fall 2008. 47 − 54. [13] Ikorong Anouk Gilbert Nemron. A Curious Strong Resemblance Between The Goldbach Conjecture And Fermat Last Assertion. Journal Of Informatics And Mathematical Sciences; Volume.1, Number.1, 2009. 75 − 80. [14] Ikorong Anouk Gilbert Nemron. Runing With The Twin Primes, The Goldbach Conjecture, The Fermat Primes Numbers, The Fermat Composite Numbers, And The Mersenne Primes; Far East Journal Of Mathematical Sciences; Volume 40, Issue 2, M ay2010, 253 − 266. [15] Ikorong Anouk Gilbert Nemron. Concerning The Goldbach Conjecture And The Fermat Last Assertion. Communication in Mathematics and Application 2011. [16] Ikorong Anouk Gilbert Nemron. A Proof Of The Goldbach Conjecture And The Strong Attachement To The Fermat’s Last Conjecture. South Asian Journal Of Mathematics; Vol1 (2); 2011, 68 − 80. [17] Ikorong Anouk Gilbert Nemron. A Complete Simple Proof Of The Fermat’s Last Conjecture; International Mathematical Forum; Vol.7 ,2012, no 20, 953 − 971. [18] Ikorong Anouk Gilbert Nemron. A Simple Proof Of The Fermat’s Last Conjecture And The

6

NEMRON

Connection With The Goldbach Conjecture. Journal Of Mathematical And Computational Sciences; Vol.2 ,2012, no 2. [19] Ikorong Anouk Gilbert Nemron. A Simple Proof Of The Sophie Germain Primes Problem Along With The Mersenne Primes Problem And Their Connection To The Fermat’s Last Conjecture. Scientia Magna; Vol8; 2012, no 2, 42 − 61. [20] J.Bamberg, G.Cairns, and D.Kilminster. The Crystalographic Restriction, permutations, and Goldbach’s conjecture. The American Mathematical Monthly. Volume 110. Number 3, (March 2003). 202 − 209. [21] H.Pogorzelski. Transtheoretic Foundations of Mathematics (General Summary of Results). Serie I: Natural Numbers. Volume IC: Goldbach Conjecture. Edited by Wang Yuan, (1984). [22] Paul Hoffman. Erd¨ os, l’homme qui n’aimait que les nombres. Editions Belin, (2000). 30 − 49. [23] Paul Hoffman. The man who loved only numbers. The story of Paul Erd¨ os and the search for mathematical truth. 1998. 30 − 49.

Centre de Calcul, d’Enseignement et de Recherche; Universite’ Pierre et Marie Curie (Paris VI); France