Bertrand s postulate Chapter 2

Size: px
Start display at page:

Download "Bertrand s postulate Chapter 2"

Transcription

1 Bertrad s postulate Chapter We have see that the sequece of prie ubers, 3, 5, 7,... is ifiite. To see that the size of its gaps is ot bouded, let N := 3 5 p deote the product of all prie ubers that are saller tha +, ad ote that oe of the ubers N +, N + 3, N + 4,...,N +, N + + is prie, sice for i + we ow that i has a prie factor that is saller tha +, ad this factor also divides N, ad hece also N + i. With this recipe, we fid, for exaple, for = 0 that oe of the te ubers 3, 33, 34,..., 3 is prie. But there are also upper bouds for the gaps i the sequece of prie ubers. A faous boud states that the gap to the ext prie caot be larger tha the uber we start our search at. This is ow as Bertrad s postulate, sice it was cojectured ad verified epirically for < by Joseph Bertrad. It was first proved for all by Pafuty Chebyshev i 850. A uch sipler proof was give by the Idia geius Raauja. Our Boo Proof is by Paul Erdős: it is tae fro Erdős first published paper, which appeared i 93, whe Erdős was 9. Joseph Bertrad Bertrad s postulate. For every, there is soe prie uber p with < p. Proof. We will estiate the size of the bioial coefficiet carefully eough to see that if it did t have ay prie factors i the rage < p, the it would be too sall. Our arguet is i five steps. We first prove Bertrad s postulate for < For this oe does ot eed to chec 4000 cases: it suffices this is Ladau s tric to chec that, 3, 5, 7, 3, 3, 43, 83,63, 37, 63, 59, 503, 400 is a sequece of prie ubers, where each is saller tha twice the previous oe. Hece every iterval {y : < y }, with 4000, cotais oe of these 4 pries.

2 8 Bertrad s postulate Next we prove that p 4 x for all real x, p x where our otatio here ad i the followig is eat to iply that the product is tae over all prie ubers p x. The proof that we preset for this fact uses iductio o the uber of these pries. It is ot fro Erdős origial paper, but it is also due to Erdős see the argi, ad it is a true Boo Proof. First we ote that if q is the largest prie with q x, the p ad 4 q 4 x. p = p x p q Thus it suffices to chec for the case where x = q is a prie uber. For q = we get 4, so we proceed to cosider odd pries q = +. Here we ay assue, by iductio, that is valid for all itegers x i the set {, 3,..., }. For q = + we split the product ad copute p = + p p 4 4 = 4. p + p + +<p + All the pieces of this oe-lie coputatio are easy to see. I fact, p 4 p + holds by iductio. The iequality +<p + p + Legedre s theore The uber! cotais the prie factor p exactly X j p ties. Proof. Exactly p of the factors of! = 3 are divisible by p, which accouts for p p-factors. Next, p of the factors of! are eve divisible by p, which accouts for the ext p prie factors p of!, etc. follows fro the observatio that + = +!!+! is a iteger, where the pries that we cosider all are factors of the uerator +!, but ot of the deoiator! +!. Fially + holds sice + ad + + are two equal! suads that appear i + + = +. =0!! co- 3 Fro Legedre s theore see the box we get that =! tais the prie factor p exactly p p

3 Bertrad s postulate 9 ties. Here each suad is at ost, sice it satisfies p p < p p =, ad it is a iteger. Furtherore the suads vaish wheever p >. Thus cotais p exactly p p ax{r : p r } ties. Hece the largest power of p that divides is ot larger tha. I particular, pries p > appear at ost oce i. Furtherore ad this, accordig to Erdős, is the ey fact for his proof pries p that satisfy 3 < p do ot divide Exaples such as at all! Ideed, `6 3p > iplies for 3, ad hece p 3 that p ad p are the oly ultiples of p that appear as factors i the uerator of!!!, while we get two p-factors i the deoiator. 4 Now we are ready to estiate. For 3, usig a estiate fro page for the lower boud, we get 4 p <p 3 p <p ad thus, sice there are ot ore tha pries p, 4 + p p for 3. <p 3 <p 5 Assue ow that there is o prie p with < p, so the secod product i is. Substitutig ito we get or , 3 which is false for large eough! I fact, usig a + < a which holds for all a, by iductio we get = < + < , 4 ad thus for 50 ad hece 8 < we obtai fro 3 ad < < 0 6 = 0/3. p 3 = `8 4 = `30 5 = illustrate that very sall prie factors p < ca appear as higher powers i `, sall pries with < 3 p appear at ost oce, while factors i the gap with < p 3 do t appear at all. This iplies /3 < 0, ad thus < 4000.

4 0 Bertrad s postulate Oe ca extract eve ore fro this type of estiates: Fro oe ca derive with the sae ethods that p 30 for 4000, <p ad thus that there are at least log 30 = 30 log + pries i the rage betwee ad. This is ot that bad a estiate: the true uber of pries i this rage is roughly / log. This follows fro the prie uber theore, which says that the liit #{p : p is prie} li / log exists, ad equals. This faous result was first proved by Hadaard ad de la Vallée-Poussi i 896; Selberg ad Erdős foud a eleetary proof without coplex aalysis tools, but still log ad ivolved i 948. O the prie uber theore itself the fial word, it sees, is still ot i: for exaple a proof of the Riea hypothesis see page 49, oe of the ajor usolved ope probles i atheatics, would also give a substatial iproveet for the estiates of the prie uber theore. But also for Bertrad s postulate, oe could expect draatic iproveets. I fact, the followig is a faous usolved proble: Is there always a prie betwee ad +? For additioal iforatio see [3, p. 9] ad [4, pp. 48, 57]. Appedix: Soe estiates Estiatig via itegrals There is a very siple-but-effective ethod of estiatig sus by itegrals as already ecoutered o page 4. For estiatig the haroic ubers H = = = we draw the figure i the argi ad derive fro it H = < dt = log t by coparig the area below the graph of ft = t t with the area of the dar shaded rectagles, ad H = > = dt = log t

5 Bertrad s postulate by coparig with the area of the large rectagles icludig the lightly shaded parts. Tae together, this yields log + < H < log +. I particular, li H, ad the order of growth of H is give by H li log =. But uch better estiates are ow see [], such as Here O ` deotes a fuctio f 6 H such that f c holds for soe = log + γ O 6, 6 costat c. where γ is Euler s costat. Estiatig factorials Stirlig s forula The sae ethod applied to yields log! = log + log log = log! < where the itegral is easily coputed: [ log t dt = t log t t log = log t dt < log!, ] = log +. Thus we get a lower estiate o!! > e log + = e e ad at the sae tie a upper estiate.! =! < e log + = e e Here a ore careful aalysis is eeded to get the asyptotics of!, as give by Stirlig s forula! π. e Ad agai there are ore precise versios available, such as! = π + e O 4. Here f g eas that f li g =. Estiatig bioial coefficiets Just fro the defiitio of the bioial coefficiets as the uber of -subsets of a -set, we ow that the sequece 0,,..., of bioial coefficiets

6 Bertrad s postulate Pascal s triagle sus to =0 = is syetric: =. Fro the fuctioal equatio = + oe easily fids that for every the bioial coefficiets for a sequece that is syetric ad uiodal: it icreases towards the iddle, so that the iddle bioial coefficiets are the largest oes i the sequece: = 0 < < < / = / > > > =. Here x resp. x deotes the uber x rouded dow resp. rouded up to the earest iteger. Fro the asyptotic forulas for the factorials etioed above oe ca obtai very precise estiates for the sizes of bioial coefficiets. However, we will oly eed very wea ad siple estiates i this boo, such as the followig: for all, while for we have /, with equality oly for =. I particular, for, 4. This holds sice /, a iddle bioial coefficiet, is the largest etry i the sequece 0 +,,,...,, whose su is, ad whose average is thus. O the other had, we ote the upper boud for bioial coefficiets + =!!, which is a reasoably good estiate for the sall bioial coefficiets at the tails of the sequece, whe is large copared to. Refereces [] P. ERDŐS: Beweis eies Satzes vo Tschebyschef, Acta Sci. Math. Szeged , [] R. L. GRAHAM, D. E. KNUTH & O. PATASHNIK: Cocrete Matheatics. A Foudatio for Coputer Sciece, Addiso-Wesley, Readig MA 989. [3] G. H. HARDY & E. M. WRIGHT: A Itroductio to the Theory of Nubers, fifth editio, Oxford Uiversity Press 979. [4] P. RIBENBOIM: The New Boo of Prie Nuber Records, Spriger-Verlag, New Yor 989.

7

Binomial transform of products

Binomial transform of products Jauary 02 207 Bioial trasfor of products Khristo N Boyadzhiev Departet of Matheatics ad Statistics Ohio Norther Uiversity Ada OH 4580 USA -boyadzhiev@ouedu Abstract Give the bioial trasfors { b } ad {

More information

x !1! + 1!2!

x !1! + 1!2! 4 Euler-Maclauri Suatio Forula 4. Beroulli Nuber & Beroulli Polyoial 4.. Defiitio of Beroulli Nuber Beroulli ubers B (,,3,) are defied as coefficiets of the followig equatio. x e x - B x! 4.. Expreesio

More information

Bernoulli Polynomials Talks given at LSBU, October and November 2015 Tony Forbes

Bernoulli Polynomials Talks given at LSBU, October and November 2015 Tony Forbes Beroulli Polyoials Tals give at LSBU, October ad Noveber 5 Toy Forbes Beroulli Polyoials The Beroulli polyoials B (x) are defied by B (x), Thus B (x) B (x) ad B (x) x, B (x) x x + 6, B (x) dx,. () B 3

More information

On The Prime Numbers In Intervals

On The Prime Numbers In Intervals O The Prie Nubers I Itervals arxiv:1706.01009v1 [ath.nt] 4 Ju 2017 Kyle D. Balliet A Thesis Preseted to the Faculty of the Departet of Matheatics West Chester Uiversity West Chester, Pesylvaia I Partial

More information

Some remarks on the paper Some elementary inequalities of G. Bennett

Some remarks on the paper Some elementary inequalities of G. Bennett Soe rears o the paper Soe eleetary iequalities of G. Beett Dag Ah Tua ad Luu Quag Bay Vieta Natioal Uiversity - Haoi Uiversity of Sciece Abstract We give soe couterexaples ad soe rears of soe of the corollaries

More information

A PROBABILITY PROBLEM

A PROBABILITY PROBLEM A PROBABILITY PROBLEM A big superarket chai has the followig policy: For every Euros you sped per buy, you ear oe poit (suppose, e.g., that = 3; i this case, if you sped 8.45 Euros, you get two poits,

More information

Jacobi symbols. p 1. Note: The Jacobi symbol does not necessarily distinguish between quadratic residues and nonresidues. That is, we could have ( a

Jacobi symbols. p 1. Note: The Jacobi symbol does not necessarily distinguish between quadratic residues and nonresidues. That is, we could have ( a Jacobi sybols efiitio Let be a odd positive iteger If 1, the Jacobi sybol : Z C is the costat fuctio 1 1 If > 1, it has a decopositio ( as ) a product of (ot ecessarily distict) pries p 1 p r The Jacobi

More information

FACTORS OF SUMS OF POWERS OF BINOMIAL COEFFICIENTS. Dedicated to the memory of Paul Erdős. 1. Introduction. n k. f n,a =

FACTORS OF SUMS OF POWERS OF BINOMIAL COEFFICIENTS. Dedicated to the memory of Paul Erdős. 1. Introduction. n k. f n,a = FACTORS OF SUMS OF POWERS OF BINOMIAL COEFFICIENTS NEIL J. CALKIN Abstract. We prove divisibility properties for sus of powers of bioial coefficiets ad of -bioial coefficiets. Dedicated to the eory of

More information

Riemann Hypothesis Proof

Riemann Hypothesis Proof Riea Hypothesis Proof H. Vic Dao 43 rd West Coast Nuber Theory Coferece, Dec 6-0, 009, Asiloar, CA. Revised 0/8/00. Riea Hypothesis Proof H. Vic Dao vic0@cocast.et March, 009 Revised Deceber, 009 Abstract

More information

Integrals of Functions of Several Variables

Integrals of Functions of Several Variables Itegrals of Fuctios of Several Variables We ofte resort to itegratios i order to deterie the exact value I of soe quatity which we are uable to evaluate by perforig a fiite uber of additio or ultiplicatio

More information

Math 2784 (or 2794W) University of Connecticut

Math 2784 (or 2794W) University of Connecticut ORDERS OF GROWTH PAT SMITH Math 2784 (or 2794W) Uiversity of Coecticut Date: Mar. 2, 22. ORDERS OF GROWTH. Itroductio Gaiig a ituitive feel for the relative growth of fuctios is importat if you really

More information

6.4 Binomial Coefficients

6.4 Binomial Coefficients 64 Bioial Coefficiets Pascal s Forula Pascal s forula, aed after the seveteeth-cetury Frech atheaticia ad philosopher Blaise Pascal, is oe of the ost faous ad useful i cobiatorics (which is the foral ter

More information

CERTAIN CONGRUENCES FOR HARMONIC NUMBERS Kotor, Montenegro

CERTAIN CONGRUENCES FOR HARMONIC NUMBERS Kotor, Montenegro MATHEMATICA MONTISNIGRI Vol XXXVIII (017) MATHEMATICS CERTAIN CONGRUENCES FOR HARMONIC NUMBERS ROMEO METROVIĆ 1 AND MIOMIR ANDJIĆ 1 Maritie Faculty Kotor, Uiversity of Moteegro 85330 Kotor, Moteegro e-ail:

More information

Bertrand s Postulate

Bertrand s Postulate Bertrad s Postulate Lola Thompso Ross Program July 3, 2009 Lola Thompso (Ross Program Bertrad s Postulate July 3, 2009 1 / 33 Bertrad s Postulate I ve said it oce ad I ll say it agai: There s always a

More information

ECE 901 Lecture 4: Estimation of Lipschitz smooth functions

ECE 901 Lecture 4: Estimation of Lipschitz smooth functions ECE 9 Lecture 4: Estiatio of Lipschitz sooth fuctios R. Nowak 5/7/29 Cosider the followig settig. Let Y f (X) + W, where X is a rado variable (r.v.) o X [, ], W is a r.v. o Y R, idepedet of X ad satisfyig

More information

Automated Proofs for Some Stirling Number Identities

Automated Proofs for Some Stirling Number Identities Autoated Proofs for Soe Stirlig Nuber Idetities Mauel Kauers ad Carste Scheider Research Istitute for Sybolic Coputatio Johaes Kepler Uiversity Altebergerstraße 69 A4040 Liz, Austria Subitted: Sep 1, 2007;

More information

Math 4707 Spring 2018 (Darij Grinberg): midterm 2 page 1. Math 4707 Spring 2018 (Darij Grinberg): midterm 2 with solutions [preliminary version]

Math 4707 Spring 2018 (Darij Grinberg): midterm 2 page 1. Math 4707 Spring 2018 (Darij Grinberg): midterm 2 with solutions [preliminary version] Math 4707 Sprig 08 Darij Griberg: idter page Math 4707 Sprig 08 Darij Griberg: idter with solutios [preliiary versio] Cotets 0.. Coutig first-eve tuples......................... 3 0.. Coutig legal paths

More information

We have also learned that, thanks to the Central Limit Theorem and the Law of Large Numbers,

We have also learned that, thanks to the Central Limit Theorem and the Law of Large Numbers, Cofidece Itervals III What we kow so far: We have see how to set cofidece itervals for the ea, or expected value, of a oral probability distributio, both whe the variace is kow (usig the stadard oral,

More information

Double Derangement Permutations

Double Derangement Permutations Ope Joural of iscrete Matheatics, 206, 6, 99-04 Published Olie April 206 i SciRes http://wwwscirporg/joural/ojd http://dxdoiorg/04236/ojd2066200 ouble erageet Perutatios Pooya aeshad, Kayar Mirzavaziri

More information

A new sequence convergent to Euler Mascheroni constant

A new sequence convergent to Euler Mascheroni constant You ad Che Joural of Iequalities ad Applicatios 08) 08:7 https://doi.org/0.86/s3660-08-670-6 R E S E A R C H Ope Access A ew sequece coverget to Euler Mascheroi costat Xu You * ad Di-Rog Che * Correspodece:

More information

THE GREATEST ORDER OF THE DIVISOR FUNCTION WITH INCREASING DIMENSION

THE GREATEST ORDER OF THE DIVISOR FUNCTION WITH INCREASING DIMENSION MATHEMATICA MONTISNIGRI Vol XXVIII (013) 17-5 THE GREATEST ORDER OF THE DIVISOR FUNCTION WITH INCREASING DIMENSION GLEB V. FEDOROV * * Mechaics ad Matheatics Faculty Moscow State Uiversity Moscow, Russia

More information

Chapter 2. Asymptotic Notation

Chapter 2. Asymptotic Notation Asyptotic Notatio 3 Chapter Asyptotic Notatio Goal : To siplify the aalysis of ruig tie by gettig rid of details which ay be affected by specific ipleetatio ad hardware. [1] The Big Oh (O-Notatio) : It

More information

Discrete Mathematics: Lectures 8 and 9 Principle of Inclusion and Exclusion Instructor: Arijit Bishnu Date: August 11 and 13, 2009

Discrete Mathematics: Lectures 8 and 9 Principle of Inclusion and Exclusion Instructor: Arijit Bishnu Date: August 11 and 13, 2009 Discrete Matheatics: Lectures 8 ad 9 Priciple of Iclusio ad Exclusio Istructor: Arijit Bishu Date: August ad 3, 009 As you ca observe by ow, we ca cout i various ways. Oe such ethod is the age-old priciple

More information

Two Topics in Number Theory: Sum of Divisors of the Factorial and a Formula for Primes

Two Topics in Number Theory: Sum of Divisors of the Factorial and a Formula for Primes Iteratioal Mathematical Forum, Vol. 2, 207, o. 9, 929-935 HIKARI Ltd, www.m-hiari.com https://doi.org/0.2988/imf.207.7088 Two Topics i Number Theory: Sum of Divisors of the Factorial ad a Formula for Primes

More information

Summer MA Lesson 13 Section 1.6, Section 1.7 (part 1)

Summer MA Lesson 13 Section 1.6, Section 1.7 (part 1) Suer MA 1500 Lesso 1 Sectio 1.6, Sectio 1.7 (part 1) I Solvig Polyoial Equatios Liear equatio ad quadratic equatios of 1 variable are specific types of polyoial equatios. Soe polyoial equatios of a higher

More information

Orthogonal Functions

Orthogonal Functions Royal Holloway Uiversity of odo Departet of Physics Orthogoal Fuctios Motivatio Aalogy with vectors You are probably failiar with the cocept of orthogoality fro vectors; two vectors are orthogoal whe they

More information

MATH 324 Summer 2006 Elementary Number Theory Solutions to Assignment 2 Due: Thursday July 27, 2006

MATH 324 Summer 2006 Elementary Number Theory Solutions to Assignment 2 Due: Thursday July 27, 2006 MATH 34 Summer 006 Elemetary Number Theory Solutios to Assigmet Due: Thursday July 7, 006 Departmet of Mathematical ad Statistical Scieces Uiversity of Alberta Questio [p 74 #6] Show that o iteger of the

More information

TR/46 OCTOBER THE ZEROS OF PARTIAL SUMS OF A MACLAURIN EXPANSION A. TALBOT

TR/46 OCTOBER THE ZEROS OF PARTIAL SUMS OF A MACLAURIN EXPANSION A. TALBOT TR/46 OCTOBER 974 THE ZEROS OF PARTIAL SUMS OF A MACLAURIN EXPANSION by A. TALBOT .. Itroductio. A problem i approximatio theory o which I have recetly worked [] required for its solutio a proof that the

More information

A Pair of Operator Summation Formulas and Their Applications

A Pair of Operator Summation Formulas and Their Applications A Pair of Operator Suatio Forulas ad Their Applicatios Tia-Xiao He 1, Leetsch C. Hsu, ad Dogsheg Yi 3 1 Departet of Matheatics ad Coputer Sciece Illiois Wesleya Uiversity Blooigto, IL 6170-900, USA Departet

More information

GENERALIZED LEGENDRE POLYNOMIALS AND RELATED SUPERCONGRUENCES

GENERALIZED LEGENDRE POLYNOMIALS AND RELATED SUPERCONGRUENCES J. Nuber Theory 0, o., 9-9. GENERALIZED LEGENDRE POLYNOMIALS AND RELATED SUPERCONGRUENCES Zhi-Hog Su School of Matheatical Scieces, Huaiyi Noral Uiversity, Huaia, Jiagsu 00, PR Chia Eail: zhihogsu@yahoo.co

More information

Primes of the form n 2 + 1

Primes of the form n 2 + 1 Itroductio Ladau s Probles are four robles i Nuber Theory cocerig rie ubers: Goldbach s Cojecture: This cojecture states that every ositive eve iteger greater tha ca be exressed as the su of two (ot ecessarily

More information

A GENERALIZED BERNSTEIN APPROXIMATION THEOREM

A GENERALIZED BERNSTEIN APPROXIMATION THEOREM Ø Ñ Å Ø Ñ Ø Ð ÈÙ Ð Ø ÓÒ DOI: 10.2478/v10127-011-0029-x Tatra Mt. Math. Publ. 49 2011, 99 109 A GENERALIZED BERNSTEIN APPROXIMATION THEOREM Miloslav Duchoň ABSTRACT. The preset paper is cocered with soe

More information

Complete Solutions to Supplementary Exercises on Infinite Series

Complete Solutions to Supplementary Exercises on Infinite Series Coplete Solutios to Suppleetary Eercises o Ifiite Series. (a) We eed to fid the su ito partial fractios gives By the cover up rule we have Therefore Let S S A / ad A B B. Covertig the suad / the by usig

More information

distinct distinct n k n k n! n n k k n 1 if k n, identical identical p j (k) p 0 if k > n n (k)

distinct distinct n k n k n! n n k k n 1 if k n, identical identical p j (k) p 0 if k > n n (k) THE TWELVEFOLD WAY FOLLOWING GIAN-CARLO ROTA How ay ways ca we distribute objects to recipiets? Equivaletly, we wat to euerate equivalece classes of fuctios f : X Y where X = ad Y = The fuctios are subject

More information

Chapter 4. Fourier Series

Chapter 4. Fourier Series Chapter 4. Fourier Series At this poit we are ready to ow cosider the caoical equatios. Cosider, for eample the heat equatio u t = u, < (4.) subject to u(, ) = si, u(, t) = u(, t) =. (4.) Here,

More information

18.S34 (FALL, 2007) GREATEST INTEGER PROBLEMS. n + n + 1 = 4n + 2.

18.S34 (FALL, 2007) GREATEST INTEGER PROBLEMS. n + n + 1 = 4n + 2. 18.S34 (FALL, 007) GREATEST INTEGER PROBLEMS Note: We use the otatio x for the greatest iteger x, eve if the origial source used the older otatio [x]. 1. (48P) If is a positive iteger, prove that + + 1

More information

Sequences of Definite Integrals, Factorials and Double Factorials

Sequences of Definite Integrals, Factorials and Double Factorials 47 6 Joural of Iteger Sequeces, Vol. 8 (5), Article 5.4.6 Sequeces of Defiite Itegrals, Factorials ad Double Factorials Thierry Daa-Picard Departmet of Applied Mathematics Jerusalem College of Techology

More information

1.2 AXIOMATIC APPROACH TO PROBABILITY AND PROPERTIES OF PROBABILITY MEASURE 1.2 AXIOMATIC APPROACH TO PROBABILITY AND

1.2 AXIOMATIC APPROACH TO PROBABILITY AND PROPERTIES OF PROBABILITY MEASURE 1.2 AXIOMATIC APPROACH TO PROBABILITY AND NTEL- robability ad Distributios MODULE 1 ROBABILITY LECTURE 2 Topics 1.2 AXIOMATIC AROACH TO ROBABILITY AND ROERTIES OF ROBABILITY MEASURE 1.2.1 Iclusio-Exclusio Forula I the followig sectio we will discuss

More information

Different kinds of Mathematical Induction

Different kinds of Mathematical Induction Differet ids of Mathematical Iductio () Mathematical Iductio Give A N, [ A (a A a A)] A N () (First) Priciple of Mathematical Iductio Let P() be a propositio (ope setece), if we put A { : N p() is true}

More information

(s)h(s) = K( s + 8 ) = 5 and one finite zero is located at z 1

(s)h(s) = K( s + 8 ) = 5 and one finite zero is located at z 1 ROOT LOCUS TECHNIQUE 93 should be desiged differetly to eet differet specificatios depedig o its area of applicatio. We have observed i Sectio 6.4 of Chapter 6, how the variatio of a sigle paraeter like

More information

x+ 2 + c p () x c p () x is an arbitrary function. ( sin x ) dx p f() d d f() dx = x dx p cosx = cos x+ 2 d p () x + x-a r (1.

x+ 2 + c p () x c p () x is an arbitrary function. ( sin x ) dx p f() d d f() dx = x dx p cosx = cos x+ 2 d p () x + x-a r (1. Super Derivative (No-iteger ties Derivative). Super Derivative ad Super Differetiatio Defitio.. p () obtaied by cotiuig aalytically the ide of the differetiatio operator of Higher Derivative of a fuctio

More information

f(1), and so, if f is continuous, f(x) = f(1)x.

f(1), and so, if f is continuous, f(x) = f(1)x. 2.2.35: Let f be a additive fuctio. i Clearly fx = fx ad therefore f x = fx for all Z+ ad x R. Hece, for ay, Z +, f = f, ad so, if f is cotiuous, fx = fx. ii Suppose that f is bouded o soe o-epty ope set.

More information

Proc. Amer. Math. Soc. 139(2011), no. 5, BINOMIAL COEFFICIENTS AND THE RING OF p-adic INTEGERS

Proc. Amer. Math. Soc. 139(2011), no. 5, BINOMIAL COEFFICIENTS AND THE RING OF p-adic INTEGERS Proc. Amer. Math. Soc. 139(2011, o. 5, 1569 1577. BINOMIAL COEFFICIENTS AND THE RING OF p-adic INTEGERS Zhi-Wei Su* ad Wei Zhag Departmet of Mathematics, Naig Uiversity Naig 210093, People s Republic of

More information

MAT1026 Calculus II Basic Convergence Tests for Series

MAT1026 Calculus II Basic Convergence Tests for Series MAT026 Calculus II Basic Covergece Tests for Series Egi MERMUT 202.03.08 Dokuz Eylül Uiversity Faculty of Sciece Departmet of Mathematics İzmir/TURKEY Cotets Mootoe Covergece Theorem 2 2 Series of Real

More information

Lecture Outline. 2 Separating Hyperplanes. 3 Banach Mazur Distance An Algorithmist s Toolkit October 22, 2009

Lecture Outline. 2 Separating Hyperplanes. 3 Banach Mazur Distance An Algorithmist s Toolkit October 22, 2009 18.409 A Algorithist s Toolkit October, 009 Lecture 1 Lecturer: Joatha Keler Scribes: Alex Levi (009) 1 Outlie Today we ll go over soe of the details fro last class ad ake precise ay details that were

More information

On twin primes associated with the Hawkins random sieve

On twin primes associated with the Hawkins random sieve Joural of Nuber Theory 9 006 84 96 wwwelsevierco/locate/jt O twi pries associated with the Hawkis rado sieve HM Bui,JPKeatig School of Matheatics, Uiversity of Bristol, Bristol, BS8 TW, UK Received 3 July

More information

Lecture 10: Bounded Linear Operators and Orthogonality in Hilbert Spaces

Lecture 10: Bounded Linear Operators and Orthogonality in Hilbert Spaces Lecture : Bouded Liear Operators ad Orthogoality i Hilbert Spaces 34 Bouded Liear Operator Let ( X, ), ( Y, ) i i be ored liear vector spaces ad { } X Y The, T is said to be bouded if a real uber c such

More information

Problem 4: Evaluate ( k ) by negating (actually un-negating) its upper index. Binomial coefficient

Problem 4: Evaluate ( k ) by negating (actually un-negating) its upper index. Binomial coefficient Problem 4: Evaluate by egatig actually u-egatig its upper idex We ow that Biomial coefficiet r { where r is a real umber, is a iteger The above defiitio ca be recast i terms of factorials i the commo case

More information

The Riemann Zeta Function

The Riemann Zeta Function Physics 6A Witer 6 The Riema Zeta Fuctio I this ote, I will sketch some of the mai properties of the Riema zeta fuctio, ζ(x). For x >, we defie ζ(x) =, x >. () x = For x, this sum diverges. However, we

More information

and each factor on the right is clearly greater than 1. which is a contradiction, so n must be prime.

and each factor on the right is clearly greater than 1. which is a contradiction, so n must be prime. MATH 324 Summer 200 Elemetary Number Theory Solutios to Assigmet 2 Due: Wedesday July 2, 200 Questio [p 74 #6] Show that o iteger of the form 3 + is a prime, other tha 2 = 3 + Solutio: If 3 + is a prime,

More information

LOWER BOUNDS FOR MOMENTS OF ζ (ρ) 1. Introduction

LOWER BOUNDS FOR MOMENTS OF ζ (ρ) 1. Introduction LOWER BOUNDS FOR MOMENTS OF ζ ρ MICAH B. MILINOVICH AND NATHAN NG Abstract. Assuig the Riea Hypothesis, we establish lower bouds for oets of the derivative of the Riea zeta-fuctio averaged over the otrivial

More information

Generating Functions and Their Applications

Generating Functions and Their Applications Geeratig Fuctios ad Their Applicatios Agustius Peter Sahaggau MIT Matheatics Departet Class of 2007 18.104 Ter Paper Fall 2006 Abstract. Geeratig fuctios have useful applicatios i ay fields of study. I

More information

SOME FINITE SIMPLE GROUPS OF LIE TYPE C n ( q) ARE UNIQUELY DETERMINED BY THEIR ELEMENT ORDERS AND THEIR ORDER

SOME FINITE SIMPLE GROUPS OF LIE TYPE C n ( q) ARE UNIQUELY DETERMINED BY THEIR ELEMENT ORDERS AND THEIR ORDER Joural of Algebra, Nuber Theory: Advaces ad Applicatios Volue, Nuber, 010, Pages 57-69 SOME FINITE SIMPLE GROUPS OF LIE TYPE C ( q) ARE UNIQUELY DETERMINED BY THEIR ELEMENT ORDERS AND THEIR ORDER School

More information

The Hypergeometric Coupon Collection Problem and its Dual

The Hypergeometric Coupon Collection Problem and its Dual Joural of Idustrial ad Systes Egieerig Vol., o., pp -7 Sprig 7 The Hypergeoetric Coupo Collectio Proble ad its Dual Sheldo M. Ross Epstei Departet of Idustrial ad Systes Egieerig, Uiversity of Souther

More information

Series with Central Binomial Coefficients, Catalan Numbers, and Harmonic Numbers

Series with Central Binomial Coefficients, Catalan Numbers, and Harmonic Numbers 3 47 6 3 Joural of Iteger Sequeces, Vol. 5 (0), Article..7 Series with Cetral Biomial Coefficiets, Catala Numbers, ad Harmoic Numbers Khristo N. Boyadzhiev Departmet of Mathematics ad Statistics Ohio Norther

More information

Sequences and Series of Functions

Sequences and Series of Functions Chapter 6 Sequeces ad Series of Fuctios 6.1. Covergece of a Sequece of Fuctios Poitwise Covergece. Defiitio 6.1. Let, for each N, fuctio f : A R be defied. If, for each x A, the sequece (f (x)) coverges

More information

Some results on the Apostol-Bernoulli and Apostol-Euler polynomials

Some results on the Apostol-Bernoulli and Apostol-Euler polynomials Soe results o the Apostol-Beroulli ad Apostol-Euler polyoials Weipig Wag a, Cagzhi Jia a Tiaig Wag a, b a Departet of Applied Matheatics, Dalia Uiversity of Techology Dalia 116024, P. R. Chia b Departet

More information

arxiv: v1 [math.nt] 26 Feb 2014

arxiv: v1 [math.nt] 26 Feb 2014 FROBENIUS NUMBERS OF PYTHAGOREAN TRIPLES BYUNG KEON GIL, JI-WOO HAN, TAE HYUN KIM, RYUN HAN KOO, BON WOO LEE, JAEHOON LEE, KYEONG SIK NAM, HYEON WOO PARK, AND POO-SUNG PARK arxiv:1402.6440v1 [ath.nt] 26

More information

MDIV. Multiple divisor functions

MDIV. Multiple divisor functions MDIV. Multiple divisor fuctios The fuctios τ k For k, defie τ k ( to be the umber of (ordered factorisatios of ito k factors, i other words, the umber of ordered k-tuples (j, j 2,..., j k with j j 2...

More information

Seunghee Ye Ma 8: Week 5 Oct 28

Seunghee Ye Ma 8: Week 5 Oct 28 Week 5 Summary I Sectio, we go over the Mea Value Theorem ad its applicatios. I Sectio 2, we will recap what we have covered so far this term. Topics Page Mea Value Theorem. Applicatios of the Mea Value

More information

A New Type of q-szász-mirakjan Operators

A New Type of q-szász-mirakjan Operators Filoat 3:8 07, 567 568 https://doi.org/0.98/fil7867c Published by Faculty of Scieces ad Matheatics, Uiversity of Niš, Serbia Available at: http://www.pf.i.ac.rs/filoat A New Type of -Szász-Miraka Operators

More information

Math 140A Elementary Analysis Homework Questions 1

Math 140A Elementary Analysis Homework Questions 1 Math 14A Elemetary Aalysis Homewor Questios 1 1 Itroductio 1.1 The Set N of Natural Numbers 1 Prove that 1 2 2 2 2 1 ( 1(2 1 for all atural umbers. 2 Prove that 3 11 (8 5 4 2 for all N. 4 (a Guess a formula

More information

Zeros of Polynomials

Zeros of Polynomials Math 160 www.timetodare.com 4.5 4.6 Zeros of Polyomials I these sectios we will study polyomials algebraically. Most of our work will be cocered with fidig the solutios of polyomial equatios of ay degree

More information

Problem. Consider the sequence a j for j N defined by the recurrence a j+1 = 2a j + j for j > 0

Problem. Consider the sequence a j for j N defined by the recurrence a j+1 = 2a j + j for j > 0 GENERATING FUNCTIONS Give a ifiite sequece a 0,a,a,, its ordiary geeratig fuctio is A : a Geeratig fuctios are ofte useful for fidig a closed forula for the eleets of a sequece, fidig a recurrece forula,

More information

arxiv: v1 [math.nt] 10 Dec 2014

arxiv: v1 [math.nt] 10 Dec 2014 A DIGITAL BINOMIAL THEOREM HIEU D. NGUYEN arxiv:42.38v [math.nt] 0 Dec 204 Abstract. We preset a triagle of coectios betwee the Sierpisi triagle, the sum-of-digits fuctio, ad the Biomial Theorem via a

More information

A talk given at Institut Camille Jordan, Université Claude Bernard Lyon-I. (Jan. 13, 2005), and University of Wisconsin at Madison (April 4, 2006).

A talk given at Institut Camille Jordan, Université Claude Bernard Lyon-I. (Jan. 13, 2005), and University of Wisconsin at Madison (April 4, 2006). A tal give at Istitut Caille Jorda, Uiversité Claude Berard Lyo-I (Ja. 13, 005, ad Uiversity of Wiscosi at Madiso (April 4, 006. SOME CURIOUS RESULTS ON BERNOULLI AND EULER POLYNOMIALS Zhi-Wei Su Departet

More information

Solutions to Final Exam Review Problems

Solutions to Final Exam Review Problems . Let f(x) 4+x. Solutios to Fial Exam Review Problems Math 5C, Witer 2007 (a) Fid the Maclauri series for f(x), ad compute its radius of covergece. Solutio. f(x) 4( ( x/4)) ( x/4) ( ) 4 4 + x. Sice the

More information

Lecture 1. January 8, 2018

Lecture 1. January 8, 2018 Lecture 1 Jauary 8, 018 1 Primes A prime umber p is a positive iteger which caot be writte as ab for some positive itegers a, b > 1. A prime p also have the property that if p ab, the p a or p b. This

More information

Product measures, Tonelli s and Fubini s theorems For use in MAT3400/4400, autumn 2014 Nadia S. Larsen. Version of 13 October 2014.

Product measures, Tonelli s and Fubini s theorems For use in MAT3400/4400, autumn 2014 Nadia S. Larsen. Version of 13 October 2014. Product measures, Toelli s ad Fubii s theorems For use i MAT3400/4400, autum 2014 Nadia S. Larse Versio of 13 October 2014. 1. Costructio of the product measure The purpose of these otes is to preset the

More information

Name Period ALGEBRA II Chapter 1B and 2A Notes Solving Inequalities and Absolute Value / Numbers and Functions

Name Period ALGEBRA II Chapter 1B and 2A Notes Solving Inequalities and Absolute Value / Numbers and Functions Nae Period ALGEBRA II Chapter B ad A Notes Solvig Iequalities ad Absolute Value / Nubers ad Fuctios SECTION.6 Itroductio to Solvig Equatios Objectives: Write ad solve a liear equatio i oe variable. Solve

More information

On the Fibonacci-like Sequences of Higher Order

On the Fibonacci-like Sequences of Higher Order Article Iteratioal Joural of oder atheatical Scieces, 05, 3(): 5-59 Iteratioal Joural of oder atheatical Scieces Joural hoepage: wwwoderscietificpressco/jourals/ijsaspx O the Fiboacci-like Sequeces of

More information

Ma/CS 6a Class 22: Power Series

Ma/CS 6a Class 22: Power Series Ma/CS 6a Class 22: Power Series By Ada Sheffer Power Series Mooial: ax i. Polyoial: a 0 + a 1 x + a 2 x 2 + + a x. Power series: A x = a 0 + a 1 x + a 2 x 2 + Also called foral power series, because we

More information

42 Dependence and Bases

42 Dependence and Bases 42 Depedece ad Bases The spa s(a) of a subset A i vector space V is a subspace of V. This spa ay be the whole vector space V (we say the A spas V). I this paragraph we study subsets A of V which spa V

More information

Available online at J. Math. Comput. Sci. 4 (2014), No. 3, ISSN:

Available online at   J. Math. Comput. Sci. 4 (2014), No. 3, ISSN: Available olie at http://scik.org J. Math. Coput. Sci. (1, No. 3, 9-5 ISSN: 197-537 ON SYMMETRICAL FUNCTIONS WITH BOUNDED BOUNDARY ROTATION FUAD. S. M. AL SARARI 1,, S. LATHA 1 Departet of Studies i Matheatics,

More information

ECE Spring Prof. David R. Jackson ECE Dept. Notes 20

ECE Spring Prof. David R. Jackson ECE Dept. Notes 20 ECE 6341 Sprig 016 Prof. David R. Jackso ECE Dept. Notes 0 1 Spherical Wave Fuctios Cosider solvig ψ + k ψ = 0 i spherical coordiates z φ θ r y x Spherical Wave Fuctios (cot.) I spherical coordiates we

More information

Bertrand s Postulate. Theorem (Bertrand s Postulate): For every positive integer n, there is a prime p satisfying n < p 2n.

Bertrand s Postulate. Theorem (Bertrand s Postulate): For every positive integer n, there is a prime p satisfying n < p 2n. Bertrad s Postulate Our goal is to prove the followig Theorem Bertrad s Postulate: For every positive iteger, there is a prime p satisfyig < p We remark that Bertrad s Postulate is true by ispectio for,,

More information

Dr. Clemens Kroll. Abstract

Dr. Clemens Kroll. Abstract Riema s Hypothesis ad Stieltjes Cojecture Riema s Hypothesis ad Stieltjes Cojecture Dr. Clemes Kroll Abstract It is show that Riema s hypothesis is true by showig that a equivalet statemet is true. Eve

More information

The Random Walk For Dummies

The Random Walk For Dummies The Radom Walk For Dummies Richard A Mote Abstract We look at the priciples goverig the oe-dimesioal discrete radom walk First we review five basic cocepts of probability theory The we cosider the Beroulli

More information

REVIEW OF CALCULUS Herman J. Bierens Pennsylvania State University (January 28, 2004) x 2., or x 1. x j. ' ' n i'1 x i well.,y 2

REVIEW OF CALCULUS Herman J. Bierens Pennsylvania State University (January 28, 2004) x 2., or x 1. x j. ' ' n i'1 x i well.,y 2 REVIEW OF CALCULUS Hera J. Bieres Pesylvaia State Uiversity (Jauary 28, 2004) 1. Suatio Let x 1,x 2,...,x e a sequece of uers. The su of these uers is usually deoted y x 1 % x 2 %...% x ' j x j, or x 1

More information

A string of not-so-obvious statements about correlation in the data. (This refers to the mechanical calculation of correlation in the data.

A string of not-so-obvious statements about correlation in the data. (This refers to the mechanical calculation of correlation in the data. STAT-UB.003 NOTES for Wedesday 0.MAY.0 We will use the file JulieApartet.tw. We ll give the regressio of Price o SqFt, show residual versus fitted plot, save residuals ad fitted. Give plot of (Resid, Price,

More information

On the transcendence of infinite sums of values of rational functions

On the transcendence of infinite sums of values of rational functions O the trascedece of ifiite sus of values of ratioal fuctios N. Saradha ad R. Tijdea Abstract P () = We ivestigate coverget sus T = Q() ad U = P (X), Q(X) Q[X], ad Q(X) has oly siple ratioal roots. = (

More information

AVERAGE MARKS SCALING

AVERAGE MARKS SCALING TERTIARY INSTITUTIONS SERVICE CENTRE Level 1, 100 Royal Street East Perth, Wester Australia 6004 Telephoe (08) 9318 8000 Facsiile (08) 95 7050 http://wwwtisceduau/ 1 Itroductio AVERAGE MARKS SCALING I

More information

On Order of a Function of Several Complex Variables Analytic in the Unit Polydisc

On Order of a Function of Several Complex Variables Analytic in the Unit Polydisc ISSN 746-7659, Eglad, UK Joural of Iforatio ad Coutig Sciece Vol 6, No 3, 0, 95-06 O Order of a Fuctio of Several Colex Variables Aalytic i the Uit Polydisc Rata Kuar Dutta + Deartet of Matheatics, Siliguri

More information

ALMOST CONVERGENCE AND SOME MATRIX TRANSFORMATIONS

ALMOST CONVERGENCE AND SOME MATRIX TRANSFORMATIONS It. J. Cotep. Math. Sci., Vol. 1, 2006, o. 1, 39-43 ALMOST CONVERGENCE AND SOME MATRIX TRANSFORMATIONS Qaaruddi ad S. A. Mohiuddie Departet of Matheatics, Aligarh Musli Uiversity Aligarh-202002, Idia sdqaar@rediffail.co,

More information

Data Analysis and Statistical Methods Statistics 651

Data Analysis and Statistical Methods Statistics 651 Data Aalysis ad Statistical Methods Statistics 651 http://www.stat.tau.edu/~suhasii/teachig.htl Suhasii Subba Rao Exaple The itroge cotet of three differet clover plats is give below. 3DOK1 3DOK5 3DOK7

More information

Appendix to Quicksort Asymptotics

Appendix to Quicksort Asymptotics Appedix to Quicksort Asymptotics James Alle Fill Departmet of Mathematical Scieces The Johs Hopkis Uiversity jimfill@jhu.edu ad http://www.mts.jhu.edu/~fill/ ad Svate Jaso Departmet of Mathematics Uppsala

More information

(A sequence also can be thought of as the list of function values attained for a function f :ℵ X, where f (n) = x n for n 1.) x 1 x N +k x N +4 x 3

(A sequence also can be thought of as the list of function values attained for a function f :ℵ X, where f (n) = x n for n 1.) x 1 x N +k x N +4 x 3 MATH 337 Sequeces Dr. Neal, WKU Let X be a metric space with distace fuctio d. We shall defie the geeral cocept of sequece ad limit i a metric space, the apply the results i particular to some special

More information

BINOMIAL COEFFICIENT HARMONIC SUM IDENTITIES ASSOCIATED TO SUPERCONGRUENCES

BINOMIAL COEFFICIENT HARMONIC SUM IDENTITIES ASSOCIATED TO SUPERCONGRUENCES #A37 INTEGERS (20) BINOMIAL COEFFICIENT HARMONIC SUM IDENTITIES ASSOCIATED TO SUPERCONGRUENCES Derot McCarthy Departet of Matheatics, Texas A&M Uiversity, Texas ccarthy@athtauedu Received: /3/, Accepted:

More information

q-fibonacci polynomials and q-catalan numbers Johann Cigler [ ] (4) I don t know who has observed this well-known fact for the first time.

q-fibonacci polynomials and q-catalan numbers Johann Cigler [ ] (4) I don t know who has observed this well-known fact for the first time. -Fiboacci polyoials ad -Catala ubers Joha Cigler The Fiboacci polyoials satisfy the recurrece F ( x s) = s x = () F ( x s) = xf ( x s) + sf ( x s) () with iitial values F ( x s ) = ad F( x s ) = These

More information

Refinements of Jensen s Inequality for Convex Functions on the Co-Ordinates in a Rectangle from the Plane

Refinements of Jensen s Inequality for Convex Functions on the Co-Ordinates in a Rectangle from the Plane Filoat 30:3 (206, 803 84 DOI 0.2298/FIL603803A Published by Faculty of Scieces ad Matheatics, Uiversity of Niš, Serbia Available at: http://www.pf.i.ac.rs/filoat Refieets of Jese s Iequality for Covex

More information

Randomized Algorithms I, Spring 2018, Department of Computer Science, University of Helsinki Homework 1: Solutions (Discussed January 25, 2018)

Randomized Algorithms I, Spring 2018, Department of Computer Science, University of Helsinki Homework 1: Solutions (Discussed January 25, 2018) Radomized Algorithms I, Sprig 08, Departmet of Computer Sciece, Uiversity of Helsiki Homework : Solutios Discussed Jauary 5, 08). Exercise.: Cosider the followig balls-ad-bi game. We start with oe black

More information

THE ARTIN CARMICHAEL PRIMITIVE ROOT PROBLEM ON AVERAGE

THE ARTIN CARMICHAEL PRIMITIVE ROOT PROBLEM ON AVERAGE THE ARTIN CARMICHAEL PRIMITIVE ROOT PROBLEM ON AVERAGE SHUGUANG LI AND CARL POMERANCE Abstract. For a atural uber, let λ() deote the order of the largest cyclic subgroup of (Z/Z). For a give iteger a,

More information

Q-BINOMIALS AND THE GREATEST COMMON DIVISOR. Keith R. Slavin 8474 SW Chevy Place, Beaverton, Oregon 97008, USA.

Q-BINOMIALS AND THE GREATEST COMMON DIVISOR. Keith R. Slavin 8474 SW Chevy Place, Beaverton, Oregon 97008, USA. INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 8 2008, #A05 Q-BINOMIALS AND THE GREATEST COMMON DIVISOR Keith R. Slavi 8474 SW Chevy Place, Beaverto, Orego 97008, USA slavi@dsl-oly.et Received:

More information

Math 113 Exam 3 Practice

Math 113 Exam 3 Practice Math Exam Practice Exam will cover.-.9. This sheet has three sectios. The first sectio will remid you about techiques ad formulas that you should kow. The secod gives a umber of practice questios for you

More information

Recurrence Relations

Recurrence Relations Recurrece Relatios Aalysis of recursive algorithms, such as: it factorial (it ) { if (==0) retur ; else retur ( * factorial(-)); } Let t be the umber of multiplicatios eeded to calculate factorial(). The

More information

Enumerative & Asymptotic Combinatorics

Enumerative & Asymptotic Combinatorics C50 Eumerative & Asymptotic Combiatorics Stirlig ad Lagrage Sprig 2003 This sectio of the otes cotais proofs of Stirlig s formula ad the Lagrage Iversio Formula. Stirlig s formula Theorem 1 (Stirlig s

More information

MA131 - Analysis 1. Workbook 9 Series III

MA131 - Analysis 1. Workbook 9 Series III MA3 - Aalysis Workbook 9 Series III Autum 004 Cotets 4.4 Series with Positive ad Negative Terms.............. 4.5 Alteratig Series.......................... 4.6 Geeral Series.............................

More information

PRACTICE FINAL/STUDY GUIDE SOLUTIONS

PRACTICE FINAL/STUDY GUIDE SOLUTIONS Last edited December 9, 03 at 4:33pm) Feel free to sed me ay feedback, icludig commets, typos, ad mathematical errors Problem Give the precise meaig of the followig statemets i) a f) L ii) a + f) L iii)

More information

Solutions to Tutorial 5 (Week 6)

Solutions to Tutorial 5 (Week 6) The Uiversity of Sydey School of Mathematics ad Statistics Solutios to Tutorial 5 (Wee 6 MATH2962: Real ad Complex Aalysis (Advaced Semester, 207 Web Page: http://www.maths.usyd.edu.au/u/ug/im/math2962/

More information

DISTANCE BETWEEN UNCERTAIN RANDOM VARIABLES

DISTANCE BETWEEN UNCERTAIN RANDOM VARIABLES MATHEMATICAL MODELLING OF ENGINEERING PROBLEMS Vol, No, 4, pp5- http://doiorg/88/ep4 DISTANCE BETWEEN UNCERTAIN RANDOM VARIABLES Yogchao Hou* ad Weicai Peg Departet of Matheatical Scieces, Chaohu Uiversity,

More information