Bookmark Author Subjects;;Summary Contents. Machine derived contents note: Divisibility. Congruences. Quadratic Reciprocity and Quadratic Forms. Some Functions of Number Theory.

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Some Diophantine Equations. Farey Fractions and Irrational Numbers. Simple Continued Fractions. Primes and Multiplicative Number Theory. Algebraic Numbers. Video thumbnail maker for mac.

Introduction to the Theory of Numbers This is an introductory course in number theory at the undergraduate level. Topics will include divisibility, greatest common divisors, the Euclidean algorithm, the Fundamental Theorem of Arithmetic, the Chinese Remainder Theorem, Hensel's Lemma, Legendre symbols, quadratic reciprocity, simple continued.

The Partition Function. The Density of Sequences of Integers. Appendices. General References. Hints. Answers. Index.Wikipedia Read associated articles:, Bookmark Work ID 6440368.

Page/Link:Page URL:HTML link:The Free Library. Retrieved May 14 2020 from1. IntroductionLet n be a natural number. By the digital sum S(n) we mean the sumof all the digits in n. For example, S(2403) = 2 + 4 + 3 = 9. The numbern is called Niven if n is a multiple of S(n).

Hence, the number 2403 isa Niven number because 2403 = 9 x 267.Niven numbers are also called Harshad numbers, a Sanskrit nameoriginally given by D. The name Niven numbers, which seemsto be more popular, first appeared in an article by Kennedy et al. 2three years after the mathematician Ivan M.

Niven stimulated the studyof such numbers following his lecture at the Fifth Annual MiamiUniversity Conference on Number Theory in 1977.There are inifinitely many Niven numbers. This fact is very easy tosee, for instance look at any power of 10, whose digital sum is 1.Numbers of digital sums 3 or 9, such as 2403 in our earlier example, arenecessarily Niven numbers too, a concequence of the well-knowncongruenceS(n) equivalent to n (mod 9). (1.1)Nonetheless, Kennedy and Cooper 3 showed that the natural densityof the Niven numbers is zero. It means that if N(x) counts the number ofNiven numbers in the interval from 1 to x, thenMATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.Hence, Niven numbers are scarce with respect to their distributionamong the natural numbers. Subsequently, De Koninck and Doyon improvedthis result by giving a more explicit lower and upper bounds for N(x),i.e. x.sup.1-epsilon much less than N(x) much less than x loglog x/log x,for any value of epsilon 0.The concept of Niven numbers has been considered as well forintegers in different bases, not merely decimal.

In fact quite a fewgeneralizations and extensions of Niven numbers have existed in theliterature. Saadatmanesh et al.

4, for instance, considered thefollowing idea.Definition 1.1. Given n, by a subdigital sum of n we mean the sumof any number of digits taken from those in n. Then n is called a superNiven number when n is divisible by all of its subdigital sums.For example, a subdigital sum of 2403 could be 2, 4, 3, or 6 = 2 +4, or 5 = 2 + 3, or 7 = 4 + 3, or 9 = 2 + 4 + 3.

Not all of thesesubdigital sums divide 2403, hence the number 2403 is not super Niven.On the other hand, the number 360 is super Niven because 360 isdivisible by 3, 6, and 9-all the subdigital sums of 360.Super Niven numbers are infinitely many (again, look at powers of10) and they form a proper subset of the Niven numbers. Saadatmanesh etal. Gave some characteristics of super Niven numbers and, at the end oftheir article, threw two open questions as a challenge. The firstconcerns with the classification of powers of 2 which are super Niven,and the second on the infinitude of the so-called pseudo-super Nivennumbers, to be explained later here.In what follows we discover that the solutions to both questionsare rather trivial.2.

Super Niven Powers of 2The notation n will be fixed to represent a natural number. Thedigital sum S(n) and subdigital sums of n are defined as before.A difficult open problem concerns with the identification of powersof 2 which are Niven numbers.

It is clear that n = 2.sup.m is Niven ifand only if S(n) is also a power of 2. Direct calculation easily givesthe first few powers of 2 which are Niven, i.e, 1, 2, 4, 8, 2.sup.9,2.sup.36, etc. Other than the ever progressing computer-generatedresults, there is no proof whether or not this list is infinite.Saadatmanesh et al. Showed that 2.sup.m cannot be a super Nivennumber for all m greater than or equal to 2.sup.40. The proof is aconsequence of another fact, that the digital sum of 2.sup.m has alower bound given by S(2.sup.m) greater than or equal to log.sup.4m - 1. To complete this result, the first challenge we take up is thefollowing open question, numbered as appeared in 4.