aa r X i v : . [ m a t h . G M ] M a y GOLDBACH’S CONJECTURE AND EULER’S φ -FUNCTION FELIX SIDOKHINE
Abstract.
In this paper we propose an alternative formulation of the binary and ternary Gold-bach conjectures as the systems of equations involving the Euler φ -function. Introduction
Goldbach’s conjecture is known as 2 n = p + q where n > p and q are primes. Asone of a possible approach to solving (or at least reformulating) the problem is also a statementregarding an equation involving φ -function (Erdos, Moser and etc.). We propose an additionalreformulation of the Goldbach conjecture as a system of equations involving φ -function.2. Arithmetical Functions
An invariant characteristic any element a belonging to a set of natural numbers N is a number ofthe prime divisors contained in its representation. Let π denote a set of all prime numbers of N . Definition 2.1. ν p : N → Z + = { n ∈ Z | n ≥ } where p is a prime acts on any element a belongingto N as follows: ν p ( a ) = α if p α || a and ν p ( a ) = 0 if p ∤ a . Definition 2.2. ν : N → Z + , where the mapping ν acts on any element a belonging to N as follows: ν ( a ) = P p ∈ π ν p ( a ). The sum P p ∈ π ν p ( a ) consists of a finite number of terms. Definition 2.3. φ : N → N , where the mapping φ acts on any element a belonging to N as follows: φ ( a ) equals the number of integers among 1 , , , a which are prime to a , [1].Let us notice some properties of these mappings: ν ( a ) = 0 ⇐⇒ a = 1 ; ν ( a ) = 1 ⇐⇒ a is prime; φ ( a ) = a − ⇐⇒ ν ( a ) = 1.3. Prime Conjectures and Euler’s Totient Function
The Bertrand Postulate.
Bertrand’s postulate: Between any two integers n and 2 n − n >
3, there is a prime.For Bertrand’s postulate it is possible the following alternative form:
Lemma 3.1.
Between any two integers n and 2 n − n >
3, there is a prime if and only if thereexists such an integer x satisfying an inequality 0 < x < n − ν ( n + x ) = 1.Due to lemma 3.1, Bertrand’s postulate is equivalent the following statement, namely: Theorem 3.1.
The equation φ ( n + x ) + 1 = n + x has an integral solution x , satisfying aninequality 0 < x < n −
2, for every integer n > n + x ) φ ( p i ) ≡ p i , where p i runs a set of all primes p ≤ [ √ n + x ] < √ n + x ≤ √ n − x , 0 < x < n −
2, for every integer n > Proof.
Let x , 0 < x < n −
2, be a solution of the equation φ ( n + x ) + 1 = n + x hence n + x is a prime. Therefore x , 0 < x < n −
2, is a solution of the system of Fermat’s consequences( n + x ) φ ( p i ) ≡ p i , where p i runs a set of all primes p ≤ [ √ n + x ] < √ n + x . Now let x ,0 < x < n −
2, is a solution of a system of consequences ( n + x ) φ ( p i ) ≡ p i where p i runs aset of all primes p ≤ [ √ n + x ] < √ n + x then, according to trial division, n + x is a prime and x , 0 < x < n −
2, be a solution of the equation φ ( n + x ) + 1 = n + x . (cid:3) Due to Tchebycheff’s theorem, [2], and theorem 3.1 the following statement is true:
Theorem 3.2.
A system of the Fermat congruences ( n + x ) φ ( p i ) ≡ p i , where p i runs a setof all primes p ≤ [ √ n + x ] < √ n + x ≤ √ n − x , 0 < x < n −
2, for everyinteger n >
3. A number of solutions of this system of congruences is equal to π (2 n − − π ( n ).An expression π (2 n − − π ( n ) has gotten purely algebraic interpretation as a number of solutions of asystem of Fermat’s congruences ( n + x ) φ ( p i ) ≡ p i where x satisfies an inequality 0 < x < n − x and varies between π ( √ n ) and π ( √ n − The Goldbach Conjecture.
The Binary Goldbach Conjecture.
The binary Goldbach conjectures: Every even integer 2 n , n >
1, is a sum of two primes.For the binary Goldbach conjecture it is possible the following alternative form:
Lemma 3.2.
Every even integer 2 n , n >
1, is a sum two primes if and only if there is such aninteger x with a condition 2 n + 1 < x < n − ν (( x − n )(4 n − x )) = 2. Proof.
Indeed, let x satisfy an inequality 2 n + 1 < x < n − ν (( x − n )(4 n − x )) = 2. Since x − n >
1, 4 n − x > x − n, n − x are primes and 2 n is a sum of two primes. Nowlet x − n, n − x be primes so 2 n + 1 < x < n −
1. It is important to note that if 2 n = p + q where p, q are primes then there is such an integer x that p, q can be expressed by x − n, n − x . (cid:3) Theorem 3.3.
Goldbach’s conjecture is true iff an equation ν (( x − n )(4 n − x )) = 2 with a condition2 n + 1 < x < n − n > Corollary 3.1.
The binary Goldbach conjecture is true iff an equation ν ( n − x ) = 2 has asolution for every integer n > ≤ x ≤ n − Proof.
According to theorem 3.3, ν (( y − n )(4 n − y )) = 2 where 2 n + 1 < y < n −
1. Making asubstitution y = 3 n − x we will get ν (( y − n )(4 n − y )) = ν ( n − x ) = 2 where 0 ≤ x ≤ n − (cid:3) Using corollary 3.1 we can represent the binary Goldbach conjecture as follows:
OLDBACH’S CONJECTURE AND EULER’S φ -FUNCTION 3 Conjecture 3.1.
A system of equations involving φ -function(1) ( φ ( n − x ) + 1 = n − xφ ( n + x ) + 1 = n + x has an integral solution for every natural number n distinct from unity.The following statement links Goldbach’s conjecture with a system of Fermat’s congruences: Theorem 3.4.
A system of equations involving φ -function(2) ( φ ( n − x ) + 1 = n − xφ ( n + x ) + 1 = n + x has an integral solution for every natural number n great than three iff for every n exists such aninteger x , 0 ≤ x ≤ n −
3, that x is a solution of a system of the Fermat congruences(3) ( ( n − x ) φ ( p i ) ≡ p i ( n + x ) φ ( q j ) ≡ q j where p i , q j run all primes p ≤ [ √ n − x ] < √ n − x ≤ √ n, q ≤ [ √ n + x ] < √ n + x ≤ √ n − Proof.
Indeed, let a system of equations have a solution for any natural number n great than three.According to corollary 3.1 for given n there is x , 0 ≤ x ≤ n − x < n − n − x , n + x are primes so x is a solution for a system of the Fermat congruences for all primes p i and q j satisfying the inequalities p i ≤ [ √ n − x ] < √ n − x , q j ≤ [ √ n + x ] < √ n + x and so on. Nowlet for any given n there is such an integer x , 0 ≤ x ≤ n − x < n − n − x ) φ ( p i ) ≡ p i , ( n + x ) φ ( q j ) ≡ q j for all primes p i and q j satisfying the inequalities p i ≤ [ √ n − x ] < √ n − x , q j ≤ [ √ n + x ] < √ n + x thenaccording to trial division n − x , n + x are primes so a system of equations has a solution and soon. (cid:3) The Ternary Goldbach Theorem.
The ternary Goldbach theorem: Every odd integer n, n >
5, is a sum of three primes.For the ternary Goldbach theorem it is possible the following alternative form:
Lemma 3.3.
Every odd integer n, n >
5, is a sum three prime numbers if and only if there are suchthe integers x, y where 0 ≤ y < x < x + y + 2 < n + 1 < x that ν (( n − x − y )(2 x − n )( n − x + y )) = 3. Proof.
Indeed, let an integer n, n >
5, be given and there are such x, y that 0 ≤ y < x < x + y + 2
Using lemma 3.3 we can represent the ternary Goldbach therem as follows:
Theorem 3.5.
A system of equations involving φ -function(4) φ ( n − x − y ) + 1 = n − x − yφ (2 x − n ) + 1 = 2 x − nφ ( n − x + y ) + 1 = n − x + y has a solution ( x, y ) in non - negative integers for every odd integer n greater than five. Proof.
Theorem 3.5 is a direct consequence of the work [3]. (cid:3)
Consider a peculiar case of the ternary Goldbach theorem, namely:
Conjecture 3.2.
A system of equations involving φ - function is complemented with a congruence(5) φ ( n − x − y ) + 1 = n − x − yφ (2 x − n ) + 1 = 2 x − nφ ( n − x + y ) + 1 = n − x + y ( n − x − y )(2 x − n ) ≡ x, y ) in non - negative integers for every odd integer n greater than five.The next statement links a peculiar case of one with the binary Goldbach conjecture: Proposition 3.1.
Every odd integer greater than five can be represented as a sum of three primesso that at least one of them is a prime number 3 if and only if every even integer greater than twocan represent as a sum of two primes.
Proof.
The proof is inductive and quite obvious. (cid:3)
Theorem 3.6.
The peculiar case of the ternary Goldbach theorem is true if and only if the binaryGoldbach conjecture is true.
Proof.
Theorem 3.6 is a direct consequence of proposition 3.1. (cid:3) “Almost all” odd integers great than five can be written as the sum of three primes so that atleast one of them is a prime number 3 (in the sense that the fraction of odd numbers which canbe so written tends towards 1). That it is quite obvious due to the works: van der Corput, 1937;Estermann, 1938; Tchudakoff, 1938 and etc. In the case of odd integers it is known every oddinteger greater than five can be represented as a sum of three primes. Helfgott, 2014.However a possibility of a presentation of the binary Goldbach conjecture as a peculiar case of theternary Goldbach theorem puts a number of the questions before circle and sieve methods as inthese methods of solving of Goldbach’s conjectures the prime numbers are deprived of personalcharacter. Here we can note a quality jump under passing from the ternary Goldbach theorem tothe binary Goldbach conjecture which scarcely can overcome by estimating sums over primes.
OLDBACH’S CONJECTURE AND EULER’S φ -FUNCTION 5 Conclusion
We have explored the possibility of reformulating certain problems about primes as the existenceof integral solutions of the systems of equations involving the Euler φ - function.We have been successful at doing so for the ternary and binary Goldbach conjectures (with binarybeing an amended system of the ternary).While solving the systems of equations involving the Euler φ - function is far from evident, somesuccess has been achieved by studying Fermat’s congruences with varying boundary conditions.Nonetheless, it offers alternative path to existing methods, which may be worth exploring. References [1] K. Chandrasekharan,
Introduction to analytic number theory . Springer, 1968.[2] W. Sierpi´nski,