and ttuu ee 2 (logn loglogn) 1 J

Size: px
Start display at page:

Download "and ttuu ee 2 (logn loglogn) 1 J"

Transcription

1 282 ON THE DENSITY OF THE ABUNDANT NUMBEBS. But, by Lemma 3, b v a(6 M )^ 2 which is a contradiction. Thus the theorem is established. 4. It will be seen that the method used in this paper leads immediately to a much better result than o (w/log 2 n) for the number of primitive abundant numbers not exceeding n. 1 shall prove in a subsequent paper that this number lies between n gci(lognloglogn)l ttuu and n ee 2 (logn loglogn) 1 J where Cj and c 2 are two absolute constants. Before closing my paper I would express my sincere gratitude to ]\Jr. H. Davenport for having so kindly aided me in my work. A THEOREM OF SYLVESTER AND SCHUR PAUL ERDOS*. The theorem in question asserts that, if n > k, then, in the set of integers n, n-\-l, n+2,..., n+k 1, there is a number containing a prime divisor greater than k. If n = k-\-l, we obtain the well-known theorem of Chebyshev. The theorem was first asserted and proved by Sylvesterf about forty-five years ago. Recently SchurJ has rediscovered and again proved the theorem. The following proof is shorter and more elementary than the previous ones. We shall not use Chebyshev's results, so that we shall also prove Chebyshev's theorem. Received 22 March, 1934; read 20 April, f J. J. Sylvester, "On arithmetical series", Messenger of Math., 21 (1892), 1-19, ; and Collected mathematical papers, 4 (1912), X J. Schur, " Einige S&tzetiberPrimzahlen mit Anwendung auf Irreduzibilit&tsfragen'', Sitzungsberichte der preussischen Akademie der Wissenschaften, Phys. Math. Klasse, 23 (1929), P. Erdds, "Beweis eines Satzes von Tschebysehef", Ada Lit. ac Sci. Regiae Universitatis Hungaricae Franscisco-Josephinae, 5 (1932),

2 A THEOREM OF SYLVESTER AND SCHUR. 283 We first express the theorem in the following form: Ifn"^ 2k, then (, J contains a prime divisor greater thank. We shall prove first the following lemma: // (,) is divisible by a power of a prime p a, then p a The expression 7 = -. T., T, r \kj {n k)\ k\ contains the prime p with the exponent W~U 2 J~L i> 2 J; where p a ^ n< p** 1. It is immediately clear that each of these a terms has the value 0 or 1. Hence the highest power of p which can divide f, J is a. I. Let n{k) denote the number of primes less than or equal to k. It is clear that, for k > 8, ir(k) < k. Hence, if f., j had no prime factors greater than k, we should have, from the lemma, (, j < n* k. On the other hand, it is evident that n \ ) n yt ~ 1 n 2 n k-\-l ( n > J. J '" 1 " A/ fl/ -L A/ M X ( 71 \ ^ -r-j <n ik, i.e. n* k < k k, which evidently does not hold when k ^ y/n. theorem for 8 < k < y/n. Thus we have proved the It may be observed that we have also proved incidentally that, if n > 2, there is always a prime number between y/n and».

3 284 P. ERDOS Since 7r(k) < $k for k > 37* (we see the validity of this proposition by considering the number of integers less than k and not divisible by 2, 3, and 5) we can prove the theorem, just as in 1, for 37 < k < nk 2. Now we consider the general case, i.e. k > r$. We suppose that it > 37. If (, J contains no prime divisor exceeding k, then (?)< n P n P n p... This inequality is an immediate consequence of the lemma about the prime power divisors of (» ). First we shall prove f that 4*> n Pi n Pk n Pi^n / V Pic"Ss v w For this purpose, we analyse the prime factors of the binomial coefficient f 2n\ (2n)\ /2i7l\ It is evident that I ) contains every prime p such that n < p < 2n, since the numerator is divisible by p and the denominator is not. Further, ( ) is divisible by any prime p such that \/n < p ^ \/{2n) for any positive integer a. For the prime p occurs in n! to the power and in (2n)! to the power 2n + r2^i r_^ since * Schur, ibid., 7. f P. Erdos, " Egy Kiirschdk f61e elemi szdmeim^leti t^tel dltalanositdsa," Math, ds Phys. Lapok, 39 (1932),

4 A THEOREM OF SYLVESTER AND SCHUB. 285 Consequently the prime p is contained in ( j to the power since Let us now denote by {x} the least integer greater than or equal to x t and put _ \n\ _ (n] _ (n\ fll ~ltj' a% ~ (22)' "" a *~l2*j» Then 1 ^ 2 ^ 3 ^ k ^ Further, a k < f k + l = ^ + 1 <2a fc+1 +l, and so, since a k and a fc+1 are integers, If now m is the first exponent for which nj2 m < 1, then a m = 1. Since 2a x ^ w, it is evident from (2) that the interval 1 <y ^n is completely covered by the intervals «,» < V < 2a m) a m _ x < y < 2a m _ 1,..., a x < y < 2a 3. It is easily seen from (2) that the interval (2) is completely covered by the intervals [#<] < V < [#(2aJ], [^-i] < V < [v / (2a wl._ 1 )] 5... for any integer k ^ 1. From all this, it follows that n Pi n Pk n P,..<( 2 ; i )(^2)-( 2 ; m ). (3) the right side being a multiple of the left.

5 286 P. ERDOS Now we show that which in combination with (3) establishes (1). We easily prove by simple arithmetic that (4) holds for any number n less than or equal to 10. Suppose now that n^lo and that (4) holds for any integer less than n. Now ( 2a A( 2a >).J 2a A<( 2a A^ (5) \a x ) \a 2 ) \a m j \ a x ) v ' which we obtain by applying (4) with n = 2d 2 1, for it is easily seen that {$(20,-1)} = 0* {I(2a 2 -l)} = a 3,... We easily obtain by induction that, for any n ^ 5, Hence, from (5), and, since 2a 1 <n+l, 2a 2 <a 1 +l, the exponent which evidently establishes (4). a 1 +2a 2 2 <t 2a x Now U p U p H p...<4: k, (V) p<k p<vk V<&k and, in particular, taking k = -y/n, n p n p n p...<^/n. P<%/n p<v^n P^y/n If now k >.n 8, then ^/k ^ ^"^n. for Z ^ 2, and so, from (1'), Up II p II ^...<4*, and so II p II p II p.-..<a k^/n. (6)

6 A THEOREM OF SYLVESTER AND SCHUB. 287 Hence (, ) < 4 fc+v ' n, and this leads, as we shall now prove, to a contradii tion. Suppose first that n > 4fc, then (tj^lr.)- Onth fik\ (2k\ 4jfe(4& l)...(3fc-fl) 4*.2* 8* since ( t ) >-^c, 8* and thus ^ < 2 fc We can easily prove by induction that 2 s * > 2k, i.e. 2 fc ~ 3 > jfc 3, for k > 20 (which does npt need any new restriction, for in this paragraph k > 37). Thus 2 2fc/3 <2 2v/n, i.e. k<3^n. But, by hypothesis, k>n$, whence n< 3 6 = 729, which means f contradiction for n > 729. Then 3. Suppose next that then If (, ) contains again no prime divisor greater than or equal to &. Since n < 4&, ( ) & < 2k, and 2 n >2n+l for» i.e.

7 288 A THEOREM OF SYLVESTER AND SCHUR. and thus 2 2^*] > 2k, it follows that (f )* < 2 6 ^k. Since ( ) 4 > 2, 2 ifc < 2 6N/fc, which is an evident contradiction for k > 576. This means that our theorem holds for n > Finally, we have to consider the case Then U But in this case, when p ^ k is a divisor of (, I, p ^ $n; for in the denominator of n \j[k! (n k)!], p <fc < w, fc occurs to the second power, consequently it must occur in the numerator to at least the third power, and so p < $n. Since \n>7$ for w>27, we have *>/{^ri)^2l ~\/n, and from this, taking k \n in (1'), we have, just as in (6), Ak 2k i.e. Ak 2k 4«fc From this we obtain r < i.e. Now we have 2^*1 > and so 4 < lk <4z 2 s /n ) or Since A; > $n, this cannot be true if n > Thus we have proved the theorem when k^l, with the exception of a finite number of other cases. The case k ^ 7 may be easily settled by a simple discussion and the other exceptions by means of tables of primes. I close by taking the opportunity of expressing my great indebtedness to Prof. L. J. Mordell for his kind assistance.

arxiv: v2 [math.nt] 21 Mar 2018

arxiv: v2 [math.nt] 21 Mar 2018 PROOF OF BERTRAND S POSTULATE FOR n 6 MANOJ VERMA arxiv:40.468v [math.nt] Mar 08 Abstract. We add a few ideas to Erdős s proof of Bertrand s Postulate to produce one using a little calculus but requiring

More information

2 MICHAEL FILASETA rather than y n (x). The polynomials z n (x) are monic polynomials with integer coecients, and y n (x) is irreducible if and only i

2 MICHAEL FILASETA rather than y n (x). The polynomials z n (x) are monic polynomials with integer coecients, and y n (x) is irreducible if and only i THE IRREDUCIBILITY OF ALL BUT FINITELY MANY BESSEL POLYNOMIALS Michael Filaseta* 1. Introduction Grosswald conjectured that the Bessel Polynomials y n (x) = nx j=0 (n + j)! 2 j (n, j)!j! xj are all irreducible

More information

On the irreducibility of certain polynomials with coefficients as products of terms in an arithmetic progression

On the irreducibility of certain polynomials with coefficients as products of terms in an arithmetic progression On the irreducibility of certain polynomials with coefficients as products of terms in an arithmetic progression Carrie E. Finch and N. Saradha 1 Introduction In 1929, Schur [9] used prime ideals in algebraic

More information

Theorems of Sylvester and Schur

Theorems of Sylvester and Schur Theorems of Sylvester and Schur T.N. Shorey An old theorem of Sylvester states that a product of consecutive positive integers each exceeding is divisible by a prime greater than. We shall give a proof

More information

Carmen s Core Concepts (Math 135)

Carmen s Core Concepts (Math 135) Carmen s Core Concepts (Math 135) Carmen Bruni University of Waterloo Week 4 1 Principle of Mathematical Induction 2 Example 3 Base Case 4 Inductive Hypothesis 5 Inductive Step When Induction Isn t Enough

More information

ASYMPTOTIC DISTRIBUTION OF THE MAXIMUM CUMULATIVE SUM OF INDEPENDENT RANDOM VARIABLES

ASYMPTOTIC DISTRIBUTION OF THE MAXIMUM CUMULATIVE SUM OF INDEPENDENT RANDOM VARIABLES ASYMPTOTIC DISTRIBUTION OF THE MAXIMUM CUMULATIVE SUM OF INDEPENDENT RANDOM VARIABLES KAI LAI CHUNG The limiting distribution of the maximum cumulative sum 1 of a sequence of independent random variables

More information

ON THE IRREDUCIBILITY OF THE GENERALIZED LAGUERRE POLYNOMIALS

ON THE IRREDUCIBILITY OF THE GENERALIZED LAGUERRE POLYNOMIALS ON THE IRREDUCIBILITY OF THE GENERALIZED LAGUERRE POLYNOMIALS M. Filaseta Mathematics Department University of South Carolina Columbia, SC 908 filaseta@math.sc.edu http://www.math.sc.edu/ filaseta/ T.-Y.

More information

ON THE NUMBER OF PAIRS OF INTEGERS WITH LEAST COMMON MULTIPLE NOT EXCEEDING x H. JAGER

ON THE NUMBER OF PAIRS OF INTEGERS WITH LEAST COMMON MULTIPLE NOT EXCEEDING x H. JAGER MATHEMATICS ON THE NUMBER OF PAIRS OF INTEGERS WITH LEAST COMMON MULTIPLE NOT EXCEEDING x BY H. JAGER (Communicated by Prof. J. PoPKEN at the meeting of March 31, 1962) In this note we shall derive an

More information

arxiv: v6 [math.nt] 16 May 2017

arxiv: v6 [math.nt] 16 May 2017 arxiv:5.038v6 [math.nt] 6 May 207 An alternative proof of the Dirichlet prime number theorem Haifeng Xu May 7, 207 Abstract Dirichlet s theorem on arithmetic progressions called as Dirichlet prime number

More information

ON REPRESENTATIVES OF SUBSETS

ON REPRESENTATIVES OF SUBSETS 26 P. HALL ON REPRESENTATIVES OF SUBSETS P. HALL - 1. Let a set S of raw things be divided into m classes of n things each in two distinct ways, (a) and (6); so that there are m (a)-classes and m (6)-classes.

More information

MATH 215 Final. M4. For all a, b in Z, a b = b a.

MATH 215 Final. M4. For all a, b in Z, a b = b a. MATH 215 Final We will assume the existence of a set Z, whose elements are called integers, along with a well-defined binary operation + on Z (called addition), a second well-defined binary operation on

More information

Homework 3, solutions

Homework 3, solutions Homework 3, solutions Problem 1. Read the proof of Proposition 1.22 (page 32) in the book. Using simialr method prove that there are infinitely many prime numbers of the form 3n 2. Solution. Note that

More information

Proof by Induction. Andreas Klappenecker

Proof by Induction. Andreas Klappenecker Proof by Induction Andreas Klappenecker 1 Motivation Induction is an axiom which allows us to prove that certain properties are true for all positive integers (or for all nonnegative integers, or all integers

More information

M381 Number Theory 2004 Page 1

M381 Number Theory 2004 Page 1 M81 Number Theory 2004 Page 1 [[ Comments are written like this. Please send me (dave@wildd.freeserve.co.uk) details of any errors you find or suggestions for improvements. ]] Question 1 20 = 2 * 10 +

More information

Mathematics 228(Q1), Assignment 2 Solutions

Mathematics 228(Q1), Assignment 2 Solutions Mathematics 228(Q1), Assignment 2 Solutions Exercise 1.(10 marks) A natural number n > 1 is said to be square free if d N with d 2 n implies d = 1. Show that n is square free if and only if n = p 1 p k

More information

Mathematical Induction Assignments

Mathematical Induction Assignments 1 Mathematical Induction Assignments Prove the Following using Principle of Mathematical induction 1) Prove that for any positive integer number n, n 3 + 2 n is divisible by 3 2) Prove that 1 3 + 2 3 +

More information

On the Equation =<#>(«+ k)

On the Equation =<#>(«+ k) MATHEMATICS OF COMPUTATION, VOLUME 26, NUMBER 11, APRIL 1972 On the Equation =(«+ k) By M. Lai and P. Gillard Abstract. The number of solutions of the equation (n) = >(n + k), for k ä 30, at intervals

More information

10x + y = xy. y = 10x x 1 > 10,

10x + y = xy. y = 10x x 1 > 10, All Problems on Prize Exam Spring 2011 Version Date: Sat Mar 5 15:12:06 EST 2011 The source for each problem is listed below when available; but even when the source is given, the formulation of the problem

More information

Proofs Not Based On POMI

Proofs Not Based On POMI s Not Based On POMI James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University February 1, 018 Outline Non POMI Based s Some Contradiction s Triangle

More information

Climbing an Infinite Ladder

Climbing an Infinite Ladder Section 5.1 Section Summary Mathematical Induction Examples of Proof by Mathematical Induction Mistaken Proofs by Mathematical Induction Guidelines for Proofs by Mathematical Induction Climbing an Infinite

More information

Almost perfect powers in consecutive integers (II)

Almost perfect powers in consecutive integers (II) Indag. Mathem., N.S., 19 (4), 649 658 December, 2008 Almost perfect powers in consecutive integers (II) by N. Saradha and T.N. Shorey School of Mathematics, Tata Institute of Fundamental Research, Homi

More information

Notes on Systems of Linear Congruences

Notes on Systems of Linear Congruences MATH 324 Summer 2012 Elementary Number Theory Notes on Systems of Linear Congruences In this note we will discuss systems of linear congruences where the moduli are all different. Definition. Given the

More information

#A2 INTEGERS 12A (2012): John Selfridge Memorial Issue ON A CONJECTURE REGARDING BALANCING WITH POWERS OF FIBONACCI NUMBERS

#A2 INTEGERS 12A (2012): John Selfridge Memorial Issue ON A CONJECTURE REGARDING BALANCING WITH POWERS OF FIBONACCI NUMBERS #A2 INTEGERS 2A (202): John Selfridge Memorial Issue ON A CONJECTURE REGARDING BALANCING WITH POWERS OF FIBONACCI NUMBERS Saúl Díaz Alvarado Facultad de Ciencias, Universidad Autónoma del Estado de México,

More information

1981] 209 Let A have property P 2 then [7(n) is the number of primes not exceeding r.] (1) Tr(n) + c l n 2/3 (log n) -2 < max k < ir(n) + c 2 n 2/3 (l

1981] 209 Let A have property P 2 then [7(n) is the number of primes not exceeding r.] (1) Tr(n) + c l n 2/3 (log n) -2 < max k < ir(n) + c 2 n 2/3 (l 208 ~ A ~ 9 ' Note that, by (4.4) and (4.5), (7.6) holds for all nonnegatíve p. Substituting from (7.6) in (6.1) and (6.2) and evaluating coefficients of xm, we obtain the following two identities. (p

More information

CSC 344 Algorithms and Complexity. Proof by Mathematical Induction

CSC 344 Algorithms and Complexity. Proof by Mathematical Induction CSC 344 Algorithms and Complexity Lecture #1 Review of Mathematical Induction Proof by Mathematical Induction Many results in mathematics are claimed true for every positive integer. Any of these results

More information

ON TRANSCENDENTAL NUMBERS GENERATED BY CERTAIN INTEGER SEQUENCES

ON TRANSCENDENTAL NUMBERS GENERATED BY CERTAIN INTEGER SEQUENCES iauliai Math. Semin., 8 (16), 2013, 6369 ON TRANSCENDENTAL NUMBERS GENERATED BY CERTAIN INTEGER SEQUENCES Soichi IKEDA, Kaneaki MATSUOKA Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku,

More information

Note that r = 0 gives the simple principle of induction. Also it can be shown that the principle of strong induction follows from simple induction.

Note that r = 0 gives the simple principle of induction. Also it can be shown that the principle of strong induction follows from simple induction. Proof by mathematical induction using a strong hypothesis Occasionally a proof by mathematical induction is made easier by using a strong hypothesis: To show P(n) [a statement form that depends on variable

More information

SOME IDENTITIES INVOLVING THE FIBONACCI POLYNOMIALS*

SOME IDENTITIES INVOLVING THE FIBONACCI POLYNOMIALS* * Yi Yuan and Wenpeeg Zhang Research Center for Basic Science, Xi'an Jiaotong University, Xi'an Shaanxi. P.R. China (Submitted June 2000-Final Revision November 2000) 1. INTROBUCTION AND RESULTS As usual,

More information

PERIODICITY OF SOME RECURRENCE SEQUENCES MODULO M

PERIODICITY OF SOME RECURRENCE SEQUENCES MODULO M INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 8 (2008), #A42 PERIODICITY OF SOME RECURRENCE SEQUENCES MODULO M Artūras Dubickas Department of Mathematics and Informatics, Vilnius University,

More information

ADVANCED PROBLEMS AND SOLUTIONS

ADVANCED PROBLEMS AND SOLUTIONS Edited by RAYMOND E. WHITNEY Please send all communications concerning ADVANCED PROBLEMS AND SOLUTIONS to RAYMOND E. WHITNEY, MATHEMATICS DEPARTMENT, LOCK HAVEN UNIVERSITY, LOCK HA PA 17745, This department

More information

THE PRIME NUMBER THEOREM FROM log»! N. LEVINSON1. During the nineteenth century attempts were made to prove the

THE PRIME NUMBER THEOREM FROM log»! N. LEVINSON1. During the nineteenth century attempts were made to prove the THE PRIME NUMBER THEOREM FROM log»! N. LEVINSON1 During the nineteenth century attempts were made to prove the prime number theorem from the formula [l, pp. 87-95] (D log pin \ + n + n LP3J + ) log p.

More information

PRIMITIVE PERIODS OF GENERALIZED FIBONACCI SEQUENCES

PRIMITIVE PERIODS OF GENERALIZED FIBONACCI SEQUENCES PRIMITIVE PERIODS O GENERALIZED IBONACCI SEQUENCES CLAUDIA SMIT and VERWER E. OGGATT, JR. San Jose State University, San Jose, California 95192 1. IWTRODUCTIOW In this paper we are concerned with the primitive

More information

On Weird and Pseudoperfect

On Weird and Pseudoperfect mathematics of computation, volume 28, number 126, april 1974, pages 617-623 On Weird and Pseudoperfect Numbers By S. J. Benkoski and P. Krdö.s Abstract. If n is a positive integer and

More information

MATH 433 Applied Algebra Lecture 4: Modular arithmetic (continued). Linear congruences.

MATH 433 Applied Algebra Lecture 4: Modular arithmetic (continued). Linear congruences. MATH 433 Applied Algebra Lecture 4: Modular arithmetic (continued). Linear congruences. Congruences Let n be a postive integer. The integers a and b are called congruent modulo n if they have the same

More information

arxiv: v2 [math.nt] 4 Jun 2016

arxiv: v2 [math.nt] 4 Jun 2016 ON THE p-adic VALUATION OF STIRLING NUMBERS OF THE FIRST KIND PAOLO LEONETTI AND CARLO SANNA arxiv:605.07424v2 [math.nt] 4 Jun 206 Abstract. For all integers n k, define H(n, k) := /(i i k ), where the

More information

SEQUENCES, MATHEMATICAL INDUCTION, AND RECURSION

SEQUENCES, MATHEMATICAL INDUCTION, AND RECURSION CHAPTER 5 SEQUENCES, MATHEMATICAL INDUCTION, AND RECURSION Copyright Cengage Learning. All rights reserved. SECTION 5.4 Strong Mathematical Induction and the Well-Ordering Principle for the Integers Copyright

More information

Know the Well-ordering principle: Any set of positive integers which has at least one element contains a smallest element.

Know the Well-ordering principle: Any set of positive integers which has at least one element contains a smallest element. The first exam will be on Monday, June 8, 202. The syllabus will be sections. and.2 in Lax, and the number theory handout found on the class web site, plus the handout on the method of successive squaring

More information

MATH 501 Discrete Mathematics. Lecture 6: Number theory. German University Cairo, Department of Media Engineering and Technology.

MATH 501 Discrete Mathematics. Lecture 6: Number theory. German University Cairo, Department of Media Engineering and Technology. MATH 501 Discrete Mathematics Lecture 6: Number theory Prof. Dr. Slim Abdennadher, slim.abdennadher@guc.edu.eg German University Cairo, Department of Media Engineering and Technology 1 Number theory Number

More information

PERFECT POLYNOMIALS OVER F p WITH p + 1 IRREDUCIBLE DIVISORS. 1. Introduction. Let p be a prime number. For a monic polynomial A F p [x] let d

PERFECT POLYNOMIALS OVER F p WITH p + 1 IRREDUCIBLE DIVISORS. 1. Introduction. Let p be a prime number. For a monic polynomial A F p [x] let d PERFECT POLYNOMIALS OVER F p WITH p + 1 IRREDUCIBLE DIVISORS L. H. GALLARDO and O. RAHAVANDRAINY Abstract. We consider, for a fixed prime number p, monic polynomials in one variable over the finite field

More information

Proof 1: Using only ch. 6 results. Since gcd(a, b) = 1, we have

Proof 1: Using only ch. 6 results. Since gcd(a, b) = 1, we have Exercise 13. Consider positive integers a, b, and c. (a) Suppose gcd(a, b) = 1. (i) Show that if a divides the product bc, then a must divide c. I give two proofs here, to illustrate the different methods.

More information

ON VALUES OF CYCLOTOMIC POLYNOMIALS. V

ON VALUES OF CYCLOTOMIC POLYNOMIALS. V Math. J. Okayama Univ. 45 (2003), 29 36 ON VALUES OF CYCLOTOMIC POLYNOMIALS. V Dedicated to emeritus professor Kazuo Kishimoto on his seventieth birthday Kaoru MOTOSE In this paper, using properties of

More information

We want to show P (n) is true for all integers

We want to show P (n) is true for all integers Generalized Induction Proof: Let P (n) be the proposition 1 + 2 + 2 2 + + 2 n = 2 n+1 1. We want to show P (n) is true for all integers n 0. Generalized Induction Example: Use generalized induction to

More information

Arithmetic Functions Evaluated at Factorials!

Arithmetic Functions Evaluated at Factorials! Arithmetic Functions Evaluated at Factorials! Dan Baczkowski (joint work with M. Filaseta, F. Luca, and O. Trifonov) (F. Luca) Fix r Q, there are a finite number of positive integers n and m for which

More information

ON INTEGERS EXPRESSIBLE BY SOME SPECIAL LINEAR FORM. 1. Introduction

ON INTEGERS EXPRESSIBLE BY SOME SPECIAL LINEAR FORM. 1. Introduction ON INTEGERS EXPRESSIBLE BY SOME SPECIAL LINEAR FORM A. DUBICKAS and A. NOVIKAS Abstract. Let E(4) be the set of positive integers expressible by the form 4M d, where M is a multiple of the product ab and

More information

PRIME NUMBERS YANKI LEKILI

PRIME NUMBERS YANKI LEKILI PRIME NUMBERS YANKI LEKILI We denote by N the set of natural numbers: 1,2,..., These are constructed using Peano axioms. We will not get into the philosophical questions related to this and simply assume

More information

#A69 INTEGERS 13 (2013) OPTIMAL PRIMITIVE SETS WITH RESTRICTED PRIMES

#A69 INTEGERS 13 (2013) OPTIMAL PRIMITIVE SETS WITH RESTRICTED PRIMES #A69 INTEGERS 3 (203) OPTIMAL PRIMITIVE SETS WITH RESTRICTED PRIMES William D. Banks Department of Mathematics, University of Missouri, Columbia, Missouri bankswd@missouri.edu Greg Martin Department of

More information

ON A PARTITION PROBLEM OF CANFIELD AND WILF. Departamento de Matemáticas, Universidad de los Andes, Bogotá, Colombia

ON A PARTITION PROBLEM OF CANFIELD AND WILF. Departamento de Matemáticas, Universidad de los Andes, Bogotá, Colombia #A11 INTEGERS 12A (2012): John Selfridge Memorial Issue ON A PARTITION PROBLEM OF CANFIELD AND WILF Željka Ljujić Departamento de Matemáticas, Universidad de los Andes, Bogotá, Colombia z.ljujic20@uniandes.edu.co

More information

PUTNAM TRAINING NUMBER THEORY. Exercises 1. Show that the sum of two consecutive primes is never twice a prime.

PUTNAM TRAINING NUMBER THEORY. Exercises 1. Show that the sum of two consecutive primes is never twice a prime. PUTNAM TRAINING NUMBER THEORY (Last updated: December 11, 2017) Remark. This is a list of exercises on Number Theory. Miguel A. Lerma Exercises 1. Show that the sum of two consecutive primes is never twice

More information

ON THE GALOIS GROUPS OF GENERALIZED LAGUERRE POLYNOMIALS

ON THE GALOIS GROUPS OF GENERALIZED LAGUERRE POLYNOMIALS ON THE GALOIS GROUPS OF GENERALIZED LAGUERRE POLYNOMIALS SHANTA LAISHRAM Abstract. For a positive integer n and a real number α, the generalized Laguerre polynomials are defined by L (α) (n + α)(n 1 +

More information

SUBSEMiGROUPS OF THE ADDITIVE POSITIVE INTEGERS 1. INTRODUCTION

SUBSEMiGROUPS OF THE ADDITIVE POSITIVE INTEGERS 1. INTRODUCTION SUBSEMiGROUPS OF THE ADDITIVE POSITIVE INTEGERS JOHNC.HIGGINS Brigham Young University, Provo, Utah 1. INTRODUCTION Many of the attempts to obtain representations for commutative and/or Archimedean semigroups

More information

417 P. ERDÖS many positive integers n for which fln) is l-th power free, i.e. fl n) is not divisible by any integral l-th power greater than 1. In fac

417 P. ERDÖS many positive integers n for which fln) is l-th power free, i.e. fl n) is not divisible by any integral l-th power greater than 1. In fac ARITHMETICAL PROPERTIES OF POLYNOMIALS P. ERDÖS*. [Extracted from the Journal of the London Mathematical Society, Vol. 28, 1953.] 1. Throughout this paper f (x) will denote a polynomial whose coefficients

More information

Solutions Manual for Homework Sets Math 401. Dr Vignon S. Oussa

Solutions Manual for Homework Sets Math 401. Dr Vignon S. Oussa 1 Solutions Manual for Homework Sets Math 401 Dr Vignon S. Oussa Solutions Homework Set 0 Math 401 Fall 2015 1. (Direct Proof) Assume that x and y are odd integers. Then there exist integers u and v such

More information

Verbatim copying and redistribution of this document. are permitted in any medium provided this notice and. the copyright notice are preserved.

Verbatim copying and redistribution of this document. are permitted in any medium provided this notice and. the copyright notice are preserved. New Results on Primes from an Old Proof of Euler s by Charles W. Neville CWN Research 55 Maplewood Ave. West Hartford, CT 06119, U.S.A. cwneville@cwnresearch.com September 25, 2002 Revision 1, April, 2003

More information

D-MATH Algebra I HS18 Prof. Rahul Pandharipande. Solution 1. Arithmetic, Zorn s Lemma.

D-MATH Algebra I HS18 Prof. Rahul Pandharipande. Solution 1. Arithmetic, Zorn s Lemma. D-MATH Algebra I HS18 Prof. Rahul Pandharipande Solution 1 Arithmetic, Zorn s Lemma. 1. (a) Using the Euclidean division, determine gcd(160, 399). (b) Find m 0, n 0 Z such that gcd(160, 399) = 160m 0 +

More information

Proofs of the infinitude of primes

Proofs of the infinitude of primes Proofs of the infinitude of primes Tomohiro Yamada Abstract In this document, I would like to give several proofs that there exist infinitely many primes. 0 Introduction It is well known that the number

More information

94 P. ERDÖS AND Á. RÉNYI 1. Generalization of the Borel-Cantelli lemma Let [X, 1,, P] be a probability space in the sense of KOLMOGOROV [6], i. e. X a

94 P. ERDÖS AND Á. RÉNYI 1. Generalization of the Borel-Cantelli lemma Let [X, 1,, P] be a probability space in the sense of KOLMOGOROV [6], i. e. X a ON CANTOR'S SERIES WITH CONVERGENT 1 By P. ERDŐS and A. RÉNYI Mathematical Institute, Eötvös Loránd University, Budapest (Received 6 April, 1959) Introduction Let {q,, ; be an arbitrary sequence of positive

More information

Sums of Squares. Bianca Homberg and Minna Liu

Sums of Squares. Bianca Homberg and Minna Liu Sums of Squares Bianca Homberg and Minna Liu June 24, 2010 Abstract For our exploration topic, we researched the sums of squares. Certain properties of numbers that can be written as the sum of two squares

More information

64 P. ERDOS AND E. G. STRAUS It is possible that (1.1) has only a finite number of solutions for fixed t and variable k > 2, but we cannot prove this

64 P. ERDOS AND E. G. STRAUS It is possible that (1.1) has only a finite number of solutions for fixed t and variable k > 2, but we cannot prove this On Products of Consecutive Integers P. ERDŐS HUNGARIAN ACADEMY OF SCIENCE BUDAPEST, HUNGARY E. G. STRAUSt UNIVERSITY OF CALIFORNIA LOS ANGELES, CALIFORNIA 1. Introduction We define A(n, k) = (n + k)!/n!

More information

Chapter 1. Introduction to prime number theory. 1.1 The Prime Number Theorem

Chapter 1. Introduction to prime number theory. 1.1 The Prime Number Theorem Chapter 1 Introduction to prime number theory 1.1 The Prime Number Theorem In the first part of this course, we focus on the theory of prime numbers. We use the following notation: we write f( g( as if

More information

arxiv: v3 [math.nt] 29 Mar 2016

arxiv: v3 [math.nt] 29 Mar 2016 arxiv:1602.06715v3 [math.nt] 29 Mar 2016 STABILITY RESULT FOR SETS WITH 3A Z n 5 VSEVOLOD F. LEV Abstract. As an easy corollary of Kneser s Theorem, if A is a subset of the elementary abelian group Z n

More information

DISTRIBUTION OF FIBONACCI AND LUCAS NUMBERS MODULO 3 k

DISTRIBUTION OF FIBONACCI AND LUCAS NUMBERS MODULO 3 k DISTRIBUTION OF FIBONACCI AND LUCAS NUMBERS MODULO 3 k RALF BUNDSCHUH AND PETER BUNDSCHUH Dedicated to Peter Shiue on the occasion of his 70th birthday Abstract. Let F 0 = 0,F 1 = 1, and F n = F n 1 +F

More information

1 4 which satisfies (2) identically in u for at least one value of the complex variable z then s c g.l.b. I ~m-1 y~~z cn, lar m- = 0 ( co < r, s < oo)

1 4 which satisfies (2) identically in u for at least one value of the complex variable z then s c g.l.b. I ~m-1 y~~z cn, lar m- = 0 ( co < r, s < oo) Nieuw Archief voor Wiskunde (3) III 13--19 (1955) FUNCTIONS WHICH ARE SYMMETRIC ABOUT SEVERAL POINTS BY PAUL ERDÖS and MICHAEL GOLOMB (Notre Dame University) (Purdue University) 1. Let 1(t) be a real-valued

More information

ADVANCED PROBLEMS AND SOLUTIONS

ADVANCED PROBLEMS AND SOLUTIONS Edited by RAYMOND E. WHITNEY Please send all communications concerning to RAYMOND E. WHITNEY, MATHEMATICS DEPARTMENT, LOCK HAVEN UNIVERSITY, LOCK HAVEN, PA 17745. This department especially welcomes problems

More information

Playing Ball with the Largest Prime Factor

Playing Ball with the Largest Prime Factor Playing Ball with the Largest Prime Factor An Introduction to Ruth-Aaron Numbers Madeleine Farris Wellesley College July 30, 2018 The Players The Players Figure: Babe Ruth Home Run Record: 714 The Players

More information

DISJUNCTIVE RADO NUMBERS FOR x 1 + x 2 + c = x 3. Dusty Sabo Department of Mathematics, Southern Oregon University, Ashland, OR USA

DISJUNCTIVE RADO NUMBERS FOR x 1 + x 2 + c = x 3. Dusty Sabo Department of Mathematics, Southern Oregon University, Ashland, OR USA INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A29 DISJUNCTIVE RADO NUMBERS FOR x 1 + x 2 + c = x 3 Dusty Sabo Department of Mathematics, Southern Oregon University, Ashland, OR

More information

GENERATING IDEALS IN SUBRINGS OF K[[X]] VIA NUMERICAL SEMIGROUPS

GENERATING IDEALS IN SUBRINGS OF K[[X]] VIA NUMERICAL SEMIGROUPS GENERATING IDEALS IN SUBRINGS OF K[[X]] VIA NUMERICAL SEMIGROUPS SCOTT T. CHAPMAN Abstract. Let K be a field and S be the numerical semigroup generated by the positive integers n 1,..., n k. We discuss

More information

About the formalization of some results by Chebyshev in number theory via the Matita ITP

About the formalization of some results by Chebyshev in number theory via the Matita ITP About the formalization of some results by Chebyshev in number theory via the Matita ITP Dipartimento di Scienze dell Informazione Mura Anteo Zamboni 7, Bologna asperti@cs.unibo.it January 19, 2009 Outline

More information

Proofs Not Based On POMI

Proofs Not Based On POMI s Not Based On POMI James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University February 12, 2018 Outline 1 Non POMI Based s 2 Some Contradiction s 3

More information

Standard forms for writing numbers

Standard forms for writing numbers Standard forms for writing numbers In order to relate the abstract mathematical descriptions of familiar number systems to the everyday descriptions of numbers by decimal expansions and similar means,

More information

SOME REMARKS ON NUMBER THEORY BY P. ERDŐS 1. Let ABSTRACT This note contains some disconnected minor remarks on number theory. (1) Iz j I=1, 1<j<co be

SOME REMARKS ON NUMBER THEORY BY P. ERDŐS 1. Let ABSTRACT This note contains some disconnected minor remarks on number theory. (1) Iz j I=1, 1<j<co be SOME REMARKS ON NUMBER THEORY BY P. ERDŐS 1. Let ABSTRACT This note contains some disconnected minor remarks on number theory. (1) Iz j I=1, 1

More information

TC10 / 3. Finite fields S. Xambó

TC10 / 3. Finite fields S. Xambó TC10 / 3. Finite fields S. Xambó The ring Construction of finite fields The Frobenius automorphism Splitting field of a polynomial Structure of the multiplicative group of a finite field Structure of the

More information

Fermat numbers and integers of the form a k + a l + p α

Fermat numbers and integers of the form a k + a l + p α ACTA ARITHMETICA * (200*) Fermat numbers and integers of the form a k + a l + p α by Yong-Gao Chen (Nanjing), Rui Feng (Nanjing) and Nicolas Templier (Montpellier) 1. Introduction. In 1849, A. de Polignac

More information

PRACTICE PROBLEMS: SET 1

PRACTICE PROBLEMS: SET 1 PRACTICE PROBLEMS: SET MATH 437/537: PROF. DRAGOS GHIOCA. Problems Problem. Let a, b N. Show that if gcd(a, b) = lcm[a, b], then a = b. Problem. Let n, k N with n. Prove that (n ) (n k ) if and only if

More information

EECS 1028 M: Discrete Mathematics for Engineers

EECS 1028 M: Discrete Mathematics for Engineers EECS 1028 M: Discrete Mathematics for Engineers Suprakash Datta Office: LAS 3043 Course page: http://www.eecs.yorku.ca/course/1028 Also on Moodle S. Datta (York Univ.) EECS 1028 W 18 1 / 32 Proofs Proofs

More information

H am ilton ia n William T. Trotter, Jr.

H am ilton ia n William T. Trotter, Jr. When the Cartesian Product of Directed Cycles is H am ilton ia n William T. Trotter, Jr. UNlVERSlN OF SOUTH CAROLlNA Paul Erdos HUNGARlAN ACADEMY OF SClENCES ABSTRACT The Cartesian product of two hamiltonian

More information

220 B. BOLLOBÁS, P. ERDŐS AND M. SIMONOVITS then every G(n, m) contains a K,,, 1 (t), where a log n t - d log 2 (1 Jc) (2) (b) Given an integer d > 1

220 B. BOLLOBÁS, P. ERDŐS AND M. SIMONOVITS then every G(n, m) contains a K,,, 1 (t), where a log n t - d log 2 (1 Jc) (2) (b) Given an integer d > 1 ON THE STRUCTURE OF EDGE GRAPHS II B. BOLLOBAS, P. ERDŐS AND M. SIMONOVITS This note is a sequel to [1]. First let us recall some of the notations. Denote by G(n, in) a graph with n vertices and m edges.

More information

AVOIDING ZERO-SUM SUBSEQUENCES OF PRESCRIBED LENGTH OVER THE INTEGERS

AVOIDING ZERO-SUM SUBSEQUENCES OF PRESCRIBED LENGTH OVER THE INTEGERS AVOIDING ZERO-SUM SUBSEQUENCES OF PRESCRIBED LENGTH OVER THE INTEGERS C. AUGSPURGER, M. MINTER, K. SHOUKRY, P. SISSOKHO, AND K. VOSS MATHEMATICS DEPARTMENT, ILLINOIS STATE UNIVERSITY NORMAL, IL 61790 4520,

More information

A METRIC RESULT CONCERNING THE APPROXIMATION OF REAL NUMBERS BY CONTINUED FRACTIONS 1. INTRODUCTION AND STATEMENT OF RESULTS

A METRIC RESULT CONCERNING THE APPROXIMATION OF REAL NUMBERS BY CONTINUED FRACTIONS 1. INTRODUCTION AND STATEMENT OF RESULTS A METRIC RESULT CONCERNING THE APPROXIMATION OF REAL NUMBERS BY CONTINUED FRACTIONS C. Eisner Institut fur Mathematik, Technische Universitat Hannover, Welfengarten 1, Hannover, Germany {Submitted October

More information

All variables a, b, n, etc are integers unless otherwise stated. Each part of a problem is worth 5 points.

All variables a, b, n, etc are integers unless otherwise stated. Each part of a problem is worth 5 points. Math 152, Problem Set 2 solutions (2018-01-24) All variables a, b, n, etc are integers unless otherwise stated. Each part of a problem is worth 5 points. 1. Let us look at the following equation: x 5 1

More information

ON THE REDUCIBILITY OF EXACT COVERING SYSTEMS

ON THE REDUCIBILITY OF EXACT COVERING SYSTEMS ON THE REDUCIBILITY OF EXACT COVERING SYSTEMS OFIR SCHNABEL arxiv:1402.3957v2 [math.co] 2 Jun 2015 Abstract. There exist irreducible exact covering systems (ECS). These are ECS which are not a proper split

More information

2 ERDOS AND NATHANSON k > 2. Then there is a partition of the set of positive kth powers In k I n > } = A, u A Z such that Waring's problem holds inde

2 ERDOS AND NATHANSON k > 2. Then there is a partition of the set of positive kth powers In k I n > } = A, u A Z such that Waring's problem holds inde JOURNAL OF NUMBER THEORY 2, - ( 88) Partitions of Bases into Disjoint Unions of Bases* PAUL ERDÓS Mathematical Institute of the Hungarian Academy of Sciences, Budapest, Hungary AND MELVYN B. NATHANSON

More information

FACTORS OF GENERALIZED FERMAT NUMBERS

FACTORS OF GENERALIZED FERMAT NUMBERS mathematics of computation volume 64, number 20 january, pages -40 FACTORS OF GENERALIZED FERMAT NUMBERS HARVEY DUBNER AND WILFRID KELLER Abstract. Generalized Fermât numbers have the form Fb

More information

Greatest Common Divisor MATH Greatest Common Divisor. Benjamin V.C. Collins, James A. Swenson MATH 2730

Greatest Common Divisor MATH Greatest Common Divisor. Benjamin V.C. Collins, James A. Swenson MATH 2730 MATH 2730 Greatest Common Divisor Benjamin V.C. Collins James A. Swenson The world s least necessary definition Definition Let a, b Z, not both zero. The largest integer d such that d a and d b is called

More information

A CONSTRUCTION OF ARITHMETIC PROGRESSION-FREE SEQUENCES AND ITS ANALYSIS

A CONSTRUCTION OF ARITHMETIC PROGRESSION-FREE SEQUENCES AND ITS ANALYSIS A CONSTRUCTION OF ARITHMETIC PROGRESSION-FREE SEQUENCES AND ITS ANALYSIS BRIAN L MILLER & CHRIS MONICO TEXAS TECH UNIVERSITY Abstract We describe a particular greedy construction of an arithmetic progression-free

More information

Theorem 1.1 (Prime Number Theorem, Hadamard, de la Vallée Poussin, 1896). let π(x) denote the number of primes x. Then x as x. log x.

Theorem 1.1 (Prime Number Theorem, Hadamard, de la Vallée Poussin, 1896). let π(x) denote the number of primes x. Then x as x. log x. Chapter 1 Introduction 1.1 The Prime Number Theorem In this course, we focus on the theory of prime numbers. We use the following notation: we write f( g( as if lim f(/g( = 1, and denote by log the natural

More information

Arithmetic Properties for a Certain Family of Knot Diagrams

Arithmetic Properties for a Certain Family of Knot Diagrams Arithmetic Properties for a Certain Family of Knot Diagrams Darrin D. Frey Department of Science and Math Cedarville University Cedarville, OH 45314 freyd@cedarville.edu and James A. Sellers Department

More information

COMPOSITIO MATHEMATICA

COMPOSITIO MATHEMATICA COMPOSITIO MATHEMATICA R. TIJDEMAN H. G. MEIJER On integers generated by a finite number of fixed primes Compositio Mathematica, tome 29, n o 3 (1974), p. 273-286

More information

Lecture Notes 1 Basic Concepts of Mathematics MATH 352

Lecture Notes 1 Basic Concepts of Mathematics MATH 352 Lecture Notes 1 Basic Concepts of Mathematics MATH 352 Ivan Avramidi New Mexico Institute of Mining and Technology Socorro, NM 87801 June 3, 2004 Author: Ivan Avramidi; File: absmath.tex; Date: June 11,

More information

Squares in products with terms in an arithmetic progression

Squares in products with terms in an arithmetic progression ACTA ARITHMETICA LXXXVI. (998) Squares in products with terms in an arithmetic progression by N. Saradha (Mumbai). Introduction. Let d, k 2, l 2, n, y be integers with gcd(n, d) =. Erdős [4] and Rigge

More information

12x + 18y = 50. 2x + v = 12. (x, v) = (6 + k, 2k), k Z.

12x + 18y = 50. 2x + v = 12. (x, v) = (6 + k, 2k), k Z. Math 3, Fall 010 Assignment 3 Solutions Exercise 1. Find all the integral solutions of the following linear diophantine equations. Be sure to justify your answers. (i) 3x + y = 7. (ii) 1x + 18y = 50. (iii)

More information

Introduction to Induction (LAMC, 10/14/07)

Introduction to Induction (LAMC, 10/14/07) Introduction to Induction (LAMC, 10/14/07) Olga Radko October 1, 007 1 Definitions The Method of Mathematical Induction (MMI) is usually stated as one of the axioms of the natural numbers (so-called Peano

More information

arxiv: v1 [math.nt] 19 Dec 2018

arxiv: v1 [math.nt] 19 Dec 2018 ON SECOND ORDER LINEAR SEQUENCES OF COMPOSITE NUMBERS DAN ISMAILESCU 2, ADRIENNE KO 1, CELINE LEE 3, AND JAE YONG PARK 4 arxiv:1812.08041v1 [math.nt] 19 Dec 2018 Abstract. In this paper we present a new

More information

Chapter 1. Introduction to prime number theory. 1.1 The Prime Number Theorem

Chapter 1. Introduction to prime number theory. 1.1 The Prime Number Theorem Chapter 1 Introduction to prime number theory 1.1 The Prime Number Theorem In the first part of this course, we focus on the theory of prime numbers. We use the following notation: we write f g as if lim

More information

MAT246H1S - Concepts In Abstract Mathematics. Solutions to Term Test 1 - February 1, 2018

MAT246H1S - Concepts In Abstract Mathematics. Solutions to Term Test 1 - February 1, 2018 MAT246H1S - Concepts In Abstract Mathematics Solutions to Term Test 1 - February 1, 2018 Time allotted: 110 minutes. Aids permitted: None. Comments: Statements of Definitions, Principles or Theorems should

More information

4- prime p > k with ak'1 = 1, but this is also intractable at the moment. It often happens in number theory that every new result suggests many new qu

4- prime p > k with ak'1 = 1, but this is also intractable at the moment. It often happens in number theory that every new result suggests many new qu CONSECUTIVE INTEGERS 3 by P. Erdos Very recently two old problems on consecutive integers were settled. Catalan conjectured that 8 and 9 are the only consecutive powers. First of all observe that four

More information

On rational numbers associated with arithmetic functions evaluated at factorials

On rational numbers associated with arithmetic functions evaluated at factorials On rational numbers associated with arithmetic functions evaluated at factorials Dan Baczkowski (joint work with M. Filaseta, F. Luca, and O. Trifonov) (F. Luca) Fix r Q, there are a finite number of positive

More information

Non-averaging subsets and non-vanishing transversals

Non-averaging subsets and non-vanishing transversals Non-averaging subsets and non-vanishing transversals Noga Alon Imre Z. Ruzsa Abstract It is shown that every set of n integers contains a subset of size Ω(n 1/6 ) in which no element is the average of

More information

Climbing an Infinite Ladder

Climbing an Infinite Ladder Section 5.1 Section Summary Mathematical Induction Examples of Proof by Mathematical Induction Mistaken Proofs by Mathematical Induction Guidelines for Proofs by Mathematical Induction Climbing an Infinite

More information

460 [Nov. u»=^tjf> v = <*"+?", (3)

460 [Nov. u»=^tjf> v = <*+?, (3) Minora Yatouta Nichiyodogawa High School, Osaka, 555-0031, Japan (Submitted September 2000-Final Revision August 2001) 1. INTRODUCTION We consider two sequences defined by the recursion relations «o =

More information

Equidivisible consecutive integers

Equidivisible consecutive integers & Equidivisible consecutive integers Ivo Düntsch Department of Computer Science Brock University St Catherines, Ontario, L2S 3A1, Canada duentsch@cosc.brocku.ca Roger B. Eggleton Department of Mathematics

More information