Velichko I.G. Asymmetrical Problem About the Lucky Tickets

Vеlichko Igor Gеorgiеvich

Candidate of Physical and Mathematical Sciences, Associate Professor,
Department of Higher Mathematics and Physics,
Tavria State Agrotechnological University
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Prosp. B. Khmelnitskogo, 18, 72310 Melitopol, Zaporozhskaya obl., Ukraine

Abstract. We consider a set of tickets with numbers from 000001 to 999999. The lucky ticket is called, in which the sum of the first three digits equals the sum of the last three digits. The problem of lucky tickets is counting their number. The generalization of the classical problem of the lucky ticket to the case when the number of terms in comparable amounts and different radix arbitrarily. To count the number of solutions of linear diophantine equations, the terms of which are bounded by a constant, we introduced a discrete analogue of the delta function, written as the definite integral. We wrote out the formula for the number of tickets in the form of multiple sums comprising administering function. We prove several auxiliary identities for the end of trigonometric sums, it is a generalization of identities. After changing the order of summation and integration we obtain the desired formula for the number of solutions of this diophantine equation. With an equal number of terms in the sum written out expression obtained previously known classical result, which is known to the author sources obtained using more sophisticated techniques of integration in the complex domain. In this paper we use the same kind of method of trigonometric sums I.M. Vinogradov, and all the arguments are on the set of real numbers. To obtain an exact expression as an integral of the oscillating function method of stationary phase estimates are obtained, containing radicals and exponents. The results of the comparisons of calculations on the exact and approximate formulas are given.

Key words: the method of trigonometric sums, method of stationary phase, happy tickets, asymptotic behavior, trigonometric identities.

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Asymmetrical Problem About the Lucky Tickets by Velichko I.G. is licensed under a Creative Commons Attribution 4.0 International License.

Citation in EnglishScience Journal of Volgograd State University. Mathematics. Physics. №3 (28) 2015 pp. 19-26 

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