The Kernel Function and Applications to the ABC Conjecture

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1 Applied Mathematical Sciences, Vol. 13, 2019, no. 7, HIKARI Ltd, The Kernel Function and Applications to the ABC Conjecture Rafael Jakimczuk División Matemática, Universidad Nacional de Luján Buenos Aires, Argentina This article is distributed under the Creative Commons by-nc-nd Attribution License. Copyright c 2019 Hikari Ltd. Abstract We prove a stronger version of a theorem on the kernel function proved by the author in a former article. Then, we obtain some theorems related with the ABC conjecture. Mathematics Subject Classification: 11A99, 11B99 Keywords: Kernel function, squareful numbers, perfect powers, squares 1 Introduction, Preinary Notes and Main Results A quadratfrei number or squarefree number is a product of distinct primes, that is, a number such that its prime factorization is of the form q 1 q s where the q i (i = 1,..., s) are the distinct primes in the prime factorization. We also consider 1 as squarefree. Let Q() be the number of squarefree not eceeding. It is well-known (see [1]) these numbers have positive density 6. That is, π 2 Q() = 6 + o(). π 2 The kernel of a positive integer n is the greatest squarefree that divides n and in this article will be denoted u(n). The following theorem can be proved using the theory of functions of comple variable. This theorem is interesting in itself. Besides, it has very interesting consequences in number theory, as we shall see in the following theorems. The author does not know if there is some elementary proof of this theorem.

2 332 Rafael Jakimczuk Theorem 1.1 Let us consider the sums a + b = c, where c, a is a square and b is also a square. The number of different values of c in these sums we denote B(). That is, B() is the number of positive integers not eceeding representable as the sum of two squares. We have B() B (1) log where B is a positive constant. Proof. See [5, Volume 2], where a better error term and the value of the positive constant B is given. Now, we establish a general theorem. Theorem 1.2 Let us consider the inequality r r n n ( 0) where the r i (i = 1,..., n) are fied positive real numbers. The number of solutions ( 1,..., n ) to this inequality, where the i (i = 1,..., n) are positive integers, will be denoted S n (). The following inequality holds n S n () 1 ( 0) n! r 1 r n Proof. If r 1 then the solutions to the inequality r 1 1 are 1 = 1,..., r 1 and consequently S1 () = r 1 r 1. On the other hand, if 0 < r 1 we have S 1 () = 0 and consequently also S 1 () r 1. Therefore the theorem is true for n = 1. Suppose that the theorem is true for n 1 1, we shall prove that the theorem is also true for n. Suppose that r r n then S n () = rn n=1 1 1 (n 1)! r 1 r n 1 S n 1 ( r n n ) 0 rn 1 1 ( r n n ) n 1 (n 1)! r 1 r n 1 rn n=1 n ( r n n ) n 1 d n = 1 n! r 1 r n Note that the function f( n ) = ( r n n ) n 1 is strictly decreasing in the interval [ ] 0, r n and in this interval the area below the function is greater than the sum of the areas of the r n rectangles of base 1 and height ( rn n ) n 1, that is, the sum rn n=1( r n n ) n 1. On the other hand, if 0 < r 1 + +r n then S n () = 0 and consequently the inequality also holds. The theorem is proved. In a previous article [4] we prove the following theorem.

3 The kernel function and applications to the ABC conjecture 333 Theorem 1.3 Let ɛ > 0 an arbitrary but fied real number. Let C ɛ () the number of positive integers c not eceeding such that u(c) c < u(c) 1+ɛ (2) C Then ɛ() (2) has density 1. = 1. That is, the set of numbers c that satisfy inequality The ABC conjecture establish that if a, b and c are positive and relatively prime integers which satisfy the equation a + b = c then for any ɛ > 0, with finitely many eceptions, we have that c = a + b < (u(abc)) 1+ɛ (3) Given ɛ > 0, Theorem 1.3 implies that almost for all c inequality c = a + b < u(a) 1+ɛ u(b) 1+ɛ u(c) 1+ɛ (4) holds, independently of a and b. This fact was observed in [4]. Note that if a, b and c are relatively prime integers then inequality (4) becomes inequality (3). Now, we shall prove a stronger version of Theorem 1.3. This stronger version is the following, Theorem 1.4 Let ɛ > 0 an arbitrary but fied real number. Let C ɛ () the number of positive integers c not eceeding such that Then u(c) c < u(c) 1+ɛ (5) C ɛ () = c ɛ () (6) where 0 c ɛ () 1 1+ɛ +o(1) = 1 ɛ 1+ɛ +o(1) and consequently C ɛ() = 1. That is, the set of numbers c that satisfy inequality (5) has density 1. Proof. The number of numbers c such that u(c) 1+ɛ c (7) will be denoted c ɛ (). Therefore the squarefree u(c) satisfy u(c) 1 1+ɛ (8) and consequently the number of these squarefree does not eceed 1 1+ɛ.

4 334 Rafael Jakimczuk Let p n be the n-th prime. Let us consider the inequality That is, the inequality p 1 p k > 1 1+ɛ (9) log (p 1 p k ) > 1 log (10) 1 + ɛ The prime number theorem is (see [1]) ϑ() = p log p. If we put = p k then we obtain (prime number theorem) log (p 1 p k ) p k k log k. Therefore inequality (10) can be written in the form f(k)k log k > 1 log (11) 1 + ɛ where k f(k) = 1. A solution to inequality (11) and consequently to inequality (9) is h log h log k = = g() 1 + ɛ log log 1 + ɛ log log (12) where g() = 1 and h > 1. Note that log k = h() log log, where h() = 1. Consequently k (see (12)) is an upper bound for the number of primes in the prime factorization of the squarefree u(c) (see (8)). Suppose that u(c) = q 1 q s (s = 1, 2,..., k) where the q i (i = 1,..., s) are distinct primes. Then the number of c not eceeding with the kernel q 1 q s is the number of solutions (r 1,..., r s ) (denoted M s ()) to the equation That is to the equation q r 1 1 q rs s ( 1) (s = 1, 2,..., k) (13) r 1 log q r s log q s log ( 1) (s = 1, 2,..., k) (14) and it has the upper bound (see Theorem 1.2) M s () 1 (log ) s s! log q 1 log q s ( 1 ) s (log ) s log 2 s! (s = 1, 2,..., k) (15) Now (use (12)) we have ( 1 ) k (log ) k log 2 k! > ( 1 ) k 1 (log ) k 1 log 2 (k 1)! > > 1 log (16) log 2

5 The kernel function and applications to the ABC conjecture 335 Therefore (use (16)) an upper bound for the number of c not eceeding with the same kernel u(c), where u(c) satisfies (8), is ( 1 ) k (log ) k log 2 k! = ( ) k 1 (log ) k = o(1) (17) log 2 k t(k)k where k t(k) = 1. Since the Stirling s formula k! 2π kk k give us e k log(k!) = t(k)k log k. Equations (8) and (17) give 0 c ɛ () 1 1+ɛ +o(1). The theorem is proved. Now, we shall prove some almost immediate consequences of Theorem 1.1 and Theorem 1.4. These consequences are related with the ABC conjecture. A number such that all primes in its prime factorization has multiplicity (eponent) greater than 1 is called squareful or powerful number. These numbers are very scarce, if A() is the number of powerful numbers not eceeding it is well known (see [2]) that A() c, where c > 1. Theorem 1.5 Let us consider the sums a + b = c, where c, a is a squareful and b is also a squareful. The number of different values of c in these sums we denote A 1 (). That is, A 1 () is the number of positive integers not eceeding representable as the sum of two squarefull. Let ɛ > 0 and let A 2 () be the number of these c not eceeding such that c < u(c) 1+ɛ. Then where 0 a 1 () 1 1+ɛ +o(1). Besides, we have A 2 () = A 1 () a 1 () (18) A 2 () A 1 () = 1 (19) Therefore, almost for all c = a + b, where a and b are squareful inequality (4) holds. Since a + b = c < u(c) 1+ɛ u(c) 1+ɛ u(b) 1+ɛ u(a) 1+ɛ (20) Proof. By Theorem 1.1 we have A 1 () B() B log, since all square is a squareful number. Equation (18) is an immediate consequence of Theorem 1.4. Therefore A 2 () A 1 () = A 1 () a 1 () A 1 () The theorem is proved. a 1 () = 1 A 1 () = = 1.

6 336 Rafael Jakimczuk A number of the form m n where m and n 2 are positive integers is called perfect power. Let N() be the number of perfect powers not eceeding. The following formula is well-known (see [3]). where f() = 1. N() = + f() 3 (21) Theorem 1.6 Let us consider the sums a+b = c, where c, a is a perfect power and b is also a perfect power. The number of different values of c in these sums we denote A 1 (). That is, A 1 () is the number of positive integers not eceeding representable as the sum of two perfect powers. Let ɛ > 0 and let A 2 () be the number of these c not eceeding such that c < u(c) 1+ɛ. We have A 1 () B (22) log where 0 a 1 () 1 1+ɛ +o(1). Besides, we have A 2 () = A 1 () a 1 () (23) A 2 () A 1 () = 1 (24) Therefore, almost for all c = a + b, where a and b are perfect powers inequality (4) holds. Since a + b = c < u(c) 1+ɛ u(c) 1+ɛ u(b) 1+ɛ u(a) 1+ɛ (25) Proof. We shall prove equation (22). The rest of the proof is trivial by Theorem 1.4. The number of squares not eceeding is, therefore by (21) the number of perfect powers not a square not eceeding is g() 3 where g() = 1. Consequently the number of numbers not eceeding that can be epressed as the sum of a square and a perfect power not a square does not eceed g() = g() 5 6 and the number of numbers not eceeding that can be epressed as the sum of two perfect power not a square does not eceed g() = g() Hence by Theorem 1.1 we obtain equation (22). Equation (23) is an immediate consequence of Theorem 1.4. The theorem is proved. Theorem 1.7 Let us consider the sums a + b = c, where c, a is a square and b is also a square. The number of different values of c in these sums we denote B() (see Theorem ). That is, B ( ) is the number of positive integers not eceeding representable as the sum of two squares. Let ɛ > 0

7 The kernel function and applications to the ABC conjecture 337 and let A 2 () be the number of these c not eceeding such that c < u(c) 1+ɛ. We have (Theorem 1.1) B() B (26) log where 0 a 1 () 1 1+ɛ +o(1). Besides, we have A 2 () = B() a 1 () (27) A 2 () B() = 1 (28) Therefore, almost for all c = a + b, where a and b are squares, inequality (4) holds. Since a + b = c < u(c) 1+ɛ u(c) 1+ɛ u(b) 1+ɛ u(a) 1+ɛ (29) Proof. The proof is the same as the proof of Theorem 1.6. It is an immediate consequence of Theorem 1.1 and Theorem 1.4. The theorem is proved. Theorem 1.8 let S 2 (N) be the number of sums a + b = c, where c N, a is a squareful number and b is not a squareful number. We have c 4 N 3 2 S2 (N) c 1 N 3 2 (30) where c 1 and c 4 are positive constants. That is, the magnitude order of S 2 (N) is N 3 2 Proof. The number of not squareful numbers not eceeding N has upper bound N and the number of squarefull not eceeding N has upper bound c 1 N. Therefore S2 (N) c 1 N 3 2. The number of not squareful numbers not eceeding N has lower bound c 2 2 N and the number of squarefull not eceeding 2 N has lower bound c N 2 3. Therefore S 2 2(N) c 4 N 3 2. The theorem is proved. Theorem 1.9 Let S 2 (N) be the number of sums a + b = c, where c N, a is a squareful number and b is not a squareful number. Let T (N) be the number of these sums that satisfy inequality (4). We have N T (N) S 2 (N) = 1 (31) That is, almost all sums of a squareful number and a not squareful number satisfy inequality (4).

8 338 Rafael Jakimczuk Proof. By Theorem 1.3 the number of c N such that c < u(c) 1+ɛ is N +o(n). Therefore for these c all sums a + b = c, in particular the sum of a squareful and a not squareful, satisfy inequality (4). For the rest of the c, whose number if o(n), the number of sums of a squareful and a not squareful that do not satisfy inequality (4) is less than or equal to o(n)o( N) = o ( N 3 2 ), since the number of squareful not eceeding N is O( N). Therefore the number of sums of a squareful and a not squareful that satisfy inequality (4) is T (N) = S 2 (N) o ( N 3 2 ) and consequently by Theorem 1.8 we have The theorem is proved. T (N) S 2 (N) = S ( ) 3 2(N) o N 2 = 1 + o(1) S 2 (N) Acknowledgements. The author is very grateful to Universidad Nacional de Luján. References [1] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Oford, [2] A. Ivić, The Riemann Zeta-Function, Dover, [3] R. Jakimczuk, On the distribution of perfect powers, Journal of Integer Sequences, 14 (2011), Article [4] R. Jakimczuk, Composite numbers IV with applications to the normal order of an arithmetical function, the kernel function and the ABC conjecture, International Mathematical Forum, 13 (2018), no. 4, [5] W. J. LeVeque, Topics in Number Theory, Volume 2, Addison Wesley, Received: February 17, 2019; Published: March 23, 2019