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Prime Divisors in the Rationality Condition for Odd Perfect Numbers
Aid#59330/Preprints/2019-09-10/www.mathjobs.org RFSC 04-01 Revised The Prime Divisors in the Rationality Condition for Odd Perfect Numbers Simon Davis Research Foundation of Southern California 8861 Villa La Jolla Drive #13595 La Jolla, CA 92037 Abstract. It is sufficient to prove that there is an excess of prime factors in the product of repunits with odd prime bases defined by the sum of divisors of the integer N = (4k + 4m+1 ℓ 2αi 1) i=1 qi to establish that there do not exist any odd integers with equality (4k+1)4m+2−1 between σ(N) and 2N. The existence of distinct prime divisors in the repunits 4k , 2α +1 Q q i −1 i , i = 1,...,ℓ, in σ(N) follows from a theorem on the primitive divisors of the Lucas qi−1 sequences and the square root of the product of 2(4k + 1), and the sequence of repunits will not be rational unless the primes are matched. Minimization of the number of prime divisors in σ(N) yields an infinite set of repunits of increasing magnitude or prime equations with no integer solutions. MSC: 11D61, 11K65 Keywords: prime divisors, rationality condition 1. Introduction While even perfect numbers were known to be given by 2p−1(2p − 1), for 2p − 1 prime, the universality of this result led to the the problem of characterizing any other possible types of perfect numbers. It was suggested initially by Descartes that it was not likely that odd integers could be perfect numbers [13]. After the work of de Bessy [3], Euler proved σ(N) that the condition = 2, where σ(N) = d|N d is the sum-of-divisors function, N d integer 4m+1 2α1 2αℓ restricted odd integers to have the form (4kP+ 1) q1 ...qℓ , with 4k + 1, q1,...,qℓ prime [18], and further, that there might exist no set of prime bases such that the perfect number condition was satisfied. -
Appendix a Tables of Fermat Numbers and Their Prime Factors
Appendix A Tables of Fermat Numbers and Their Prime Factors The problem of distinguishing prime numbers from composite numbers and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic. Carl Friedrich Gauss Disquisitiones arithmeticae, Sec. 329 Fermat Numbers Fo =3, FI =5, F2 =17, F3 =257, F4 =65537, F5 =4294967297, F6 =18446744073709551617, F7 =340282366920938463463374607431768211457, Fs =115792089237316195423570985008687907853 269984665640564039457584007913129639937, Fg =134078079299425970995740249982058461274 793658205923933777235614437217640300735 469768018742981669034276900318581864860 50853753882811946569946433649006084097, FlO =179769313486231590772930519078902473361 797697894230657273430081157732675805500 963132708477322407536021120113879871393 357658789768814416622492847430639474124 377767893424865485276302219601246094119 453082952085005768838150682342462881473 913110540827237163350510684586298239947 245938479716304835356329624224137217. The only known Fermat primes are Fo, ... , F4 • 208 17 lectures on Fermat numbers Completely Factored Composite Fermat Numbers m prime factor year discoverer 5 641 1732 Euler 5 6700417 1732 Euler 6 274177 1855 Clausen 6 67280421310721* 1855 Clausen 7 59649589127497217 1970 Morrison, Brillhart 7 5704689200685129054721 1970 Morrison, Brillhart 8 1238926361552897 1980 Brent, Pollard 8 p**62 1980 Brent, Pollard 9 2424833 1903 Western 9 P49 1990 Lenstra, Lenstra, Jr., Manasse, Pollard 9 p***99 1990 Lenstra, Lenstra, Jr., Manasse, Pollard -
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Science Horizon Volume 6 Issue 4 April, 2021 President, Odisha Bigyan Academy Editorial Board Sri Umakant Swain Prof. Niranjan Barik Prof. Ramesh Chandra Parida Editor Er. Mayadhar Swain Dr. Choudhury Satyabrata Nanda Dr. Rajballav Mohanty Managing Editor Dr. Nilambar Biswal Er. Bhagat Charan Mohanty Language Expert Secretary, Odisha Bigyan Academy Prof. Sachidananda Tripathy HJCONTENTS HJ Subject Author Page 1. Editorial : Uttarakhand Glacial Burst-A Warning Er. Mayadhar Swain 178 2. The Star World Prof. B.B. Swain 180 3. Role of Physics in Information and Communication Dr. Sadasiva Biswal 186 Technology 4. National Mathematics Day, National Mathematics Binapani Saikrupa 188 Park & Srinivasa Ramanujan 5. A Brief History of Indian Math-Magicians (Part IV) Purusottam Sahoo 192 6. Glacier Dr. Manas Ranjan Senapati 197 7. Do Plants Recognise their Kins? Dr. Taranisen Panda 198 Dr. Raj Ballav Mohanty 8. Paris Climate Agreement and India’s Response Dr. Sundara Narayana Patro 201 9. Fluoride Contamination in Groundwater with Jayant Kumar Sahoo 205 Special Reference to Odisha 10. Electricity from Trees Dr. Ramesh Chandra Parida 209 11. Hydrogen as a Renewable Energy for Sustainable Prof. (Dr.) L.M. Das 212 Future 12. Nomophobia Dr. Namita Kumari Das 217 13. Quiz: Mammals Dr. Kedareswar Pradhan 220 14. Recent News on Science & Technology 222 The Cover Page depicts : Green Hydrogen Cover Design : Kalakar Sahoo APRIL, 2021 // EDITORIAL // UTTARAKHAND GLACIAL BURST- A WARNING Er. Mayadhar Swain On 7th February 2021, a flash flood occurred in Chamoli district of Uttarakhand villagers, mostly women, were uprooted in causing devastating damage and killing 204 this flood. At Reni, Risiganga joins the river persons. -
Ramanujan and Labos Primes, Their Generalizations, and Classifications
1 2 Journal of Integer Sequences, Vol. 15 (2012), 3 Article 12.1.1 47 6 23 11 Ramanujan and Labos Primes, Their Generalizations, and Classifications of Primes Vladimir Shevelev Department of Mathematics Ben-Gurion University of the Negev Beer-Sheva 84105 Israel [email protected] Abstract We study the parallel properties of the Ramanujan primes and a symmetric counter- part, the Labos primes. Further, we study all primes with these properties (generalized Ramanujan and Labos primes) and construct two kinds of sieves for them. Finally, we give a further natural generalization of these constructions and pose some conjectures and open problems. 1 Introduction The very well-known Bertrand postulate (1845) states that, for every x > 1, there exists a prime in the interval (x, 2x). This postulate quickly became a theorem when, in 1851, it was unexpectedly proved by Chebyshev (for a very elegant version of his proof, see Theorem 9.2 in [8]). In 1919, Ramanujan ([6]; [7, pp. 208–209]) gave a quite new and very simple proof of Bertrand’s postulate. Moreover, in his proof of a generalization of Bertrand’s, the following sequence of primes appeared ([11], A104272) 2, 11, 17, 29, 41, 47, 59, 67, 71, 97, 101, 107, 127, 149, 151, 167,... (1) Definition 1. For n ≥ 1, the n-th Ramanujan prime is the largest prime (Rn) for which π(Rn) − π(Rn/2) = n. Let us show that, equivalently, Rn is the smallest positive integer g(n) with the property that, if x ≥ g(n), then π(x) − π(x/2) ≥ n. -
Ramanujan Primes: Bounds, Runs, Twins, and Gaps
Ramanujan Primes: Bounds, Runs, Twins, and Gaps Jonathan Sondow 209 West 97th Street New York, NY 10025 USA [email protected] John W. Nicholson P. O. Box 2423 Arlington, TX 76004 USA [email protected] Tony D. Noe 14025 NW Harvest Lane Portland, OR 97229 USA [email protected] arXiv:1105.2249v2 [math.NT] 1 Aug 2011 Abstract The nth Ramanujan prime is the smallest positive integer Rn such that if x ≥ Rn, 1 then the interval 2 x,x contains at least n primes. We sharpen Laishram’s theorem that Rn < p3n by proving that the maximum of Rn/p3n is R5/p15 = 41/47. We give statistics on the length of the longest run of Ramanujan primes among all primes p < 10n, for n ≤ 9. We prove that if an upper twin prime is Ramanujan, then so is the lower; a table gives the number of twin primes below 10n of three types. Finally, we relate runs of Ramanujan primes to prime gaps. Along the way we state several conjectures and open problems. The Appendix explains Noe’s fast algorithm for computing R1,R2,...,Rn. 1 1 Introduction For n ≥ 1, the nth Ramanujan prime is defined as the smallest positive integer Rn with 1 the property that for any x ≥ Rn, there are at least n primes p with 2 x<p ≤ x. By its 1 minimality, Rn is indeed a prime, and the interval 2 Rn, Rn contains exactly n primes [10]. In 1919 Ramanujan proved a result which implies that Rn exists, and he gave the first five Ramanujan primes. -
Numbers Quantification
Concept Map Integers Binary System Indices & Logarithms Primes & factorisation Encryption & Computer Cryptography Real Life Applications 1.1 Introduction: In this modern age of computers, primes play an important role to make our communications secure. So understanding primes are very important. In Section 1.1., we give an introduction to primes and its properties and give its application to encryption in 1.2. The concept of binary numbers and writing a number in binary and converting a number in binary to decimal is presented in 1.3. The Section 1.4 introduces the concept of complex numbers with some basic properties. In Sections 1.5, 1.6 and 1.7. logarithms, its properties and applications to real life problems are introduced. Finally in Section 1.8 we give a number of word problems which we come across in everyday life. [2] 1.2 Prime Numbers We say integer a divides b, and write a|b in symbols, if b = ac with c ∈ Z. For example 9|27. If a|b, we say a is a factor or a is divisor of b. And we also say b is a multiple of a. It is easy to see that 1 and –1 are divisors of any integer and every non-zero integer is a divisor of 0. We write a|b if a does not divide b. For example 5|13. Prime numbers are the building blocks of number system. A positive integer p > 1 is called a prime if 1 and p are the only positive divisors of p. For example, 2, 3, 5, 7, 11, 13, .... -
1 Introduction
Received: 09.11.2016 Year: 2017, Pages: 13-26 Published: 30.11.2017 Original Article** The Abundancy Index of Divisors of Spoof Odd Perfect Numbers Jose Arnaldo1;* <[email protected]> Bebita Dris2 <[email protected]> Department of Mathematics and Physics, Far Eastern University, Manila, Philippines Abstaract − We call n a spoof odd perfect number if n is odd and n = km for two integers k; m > 1 such that σ(k)(m + 1) = 2n, where σ is the sum-of-divisors function. In this paper, we show how results analogous to those of odd perfect numbers could be established for spoof odd perfect numbers (otherwise known in the literature as Descartes numbers). In particular, we predict that an analogue of the Descartes-Frenicle-Sorli conjecture for odd perfect numbers holds for spoof odd perfect numbers. Furthermore, we conjecture that the quasi-Euler prime of a spoof odd perfect number is also its largest quasi-prime factor. Keywords − Abundancy index, spoof odd perfect number, Descartes number. 1 Introduction If N is a positive integer, then we write σ(N) for the sum of the divisors of N. A number N is perfect if σ(N) = 2N. A number M is almost perfect if σ(M) = 2M − 1. It is currently unknown whether there are innitely many even perfect numbers, or whether any odd perfect numbers (OPNs) exist. On the other hand, it is known that powers of two (including 1) are almost perfect, although it is still an open problem to rule out other possible forms of even almost perfect numbers, and also, if 1 will be the only odd almost perfect number to be discovered". -
Vedic Mathematics Indian Mathematics from Vedic Period Until
Vedic Mathematics Indian Mathematics from Vedic Period until today is ‘Vedic Mathematics’ Ravi Kumar Iyer Mob. +91 8076 4843 56 Vedic Mathematics I am sorry I am not able to meet you Physically. But once the pandemic is over, let us meet and learn VM properly. Today is only a TRILER I need your cooperation If possible pl sit with your school going children above the age of 12. They pick up very quickly I have conducted VM workshops in many leading universities in USA, Canada, Holland, Norway, Australia, New Zealand etc. Also in Royal Society My 5 Sessions on VM in Radio Sydney won maximum attendance award [email protected], www.hindugenius.blogspot.com Quotes on Indian Mathematics We owe a lot to Indians, who taught us how to count, without which no worthwhile scientific discovery could have been made. Albert Einstein Let Noble Thoughts come from all 4 directions. Rig Veda Ancient Vedic Shloka over 5,000 years back "Om purna mada purna midam Purnaat purnam udachyate Purnasya purnam adaaya Purnam eva vasishyate Om shanti shanti shantih” (Isha Upanishad) Guillaume de l'Hôpital Which translates into: 1661- 1704 France, Paris INFINITY ÷÷ INFINITY = INFINITY "That is the whole, this is the Whole; from the Whole, the Whole arises; taking away the Whole from the Whole, the Whole remains“ (Replace Whole by Infinity) Great Mathematicians of Vedic Period Indian Mathematics from Vedic Period until today is ‘Vedic Mathematics’ How old is Vedic Civilization ?? How old is Vedic Civilization ?? Thomas Alva Edison (1847 – 1931) Gramaphone 1877 Max Müller (1823 – 1900) I worship Agni who is the priest, the one who leads us from the front, who is the deity subject matter of a ritual, a yajna who is the one who makes the formal invocations in the yajna who is the source, storehouse and the bestower of all wealth, gems, precious stones etc 1 -1-1 of Rigvedam. -
Existence of Ramanujan Primes for GL(3)
Existence of Ramanujan primes for GL(3) Dinakar Ramakrishnan 253-37 Caltech, Pasadena, CA 91125 To Joe Shalika with admiration Introduction Let π beacuspformonGL(n)/Q , i.e., a cuspidal automophic representation of GL(n, A), where A denotes the adele ring of Q . We will say that a prime p is a Ramanujan prime for π iff the corresponding πp is tempered.The local component πp will necessarily be unramified for almost all p, determined by an unordered n-tuple {α1,p,α2,p,...,αn,p} of non-zero complex numbers, often represented by the corresponding diagonal matrix Ap(π)inGL(n, C), unique up to permutation of the diagonal entries. The L-factor of π at p is given by n −s −1 −s −1 L(s, πp)=det(I − Ap(π)p ) = (1 − αj,pp ) . j=1 As π is unitary, πp is tempered (in the unramified case) iff each αj,p is of absolute value 1. It was shown in [Ra] that for n = 2,thesetR(π)of Ramanujan primes for π is of lower density at least 9/10. When one applies in addition the deep recent results of H. Kim and F. Shahidi ([KSh]) on the symmetric cube and the symmetric fourth power liftings for GL(2), the lower bound improves from 9/10 to 34/35 (loc. cit.), which is 0.971428 .... For n>2 there is a dearth of results for general π, though for cusp forms of regular infinity type, assumed for n>3 to have a discrete series component at a finite place, one knows by [Pic] for n = 3, which relies on the works of J. -
Subject Index
Subject Index Many of these terms are defined in the glossary, others are defined in the Prime Curios! themselves. The boldfaced entries should indicate the key entries. γ 97 Arecibo Message 55 φ 79, 184, see golden ratio arithmetic progression 34, 81, π 8, 12, 90, 102, 106, 129, 136, 104, 112, 137, 158, 205, 210, 154, 164, 172, 173, 177, 181, 214, 219, 223, 226, 227, 236 187, 218, 230, 232, 235 Armstrong number 215 5TP39 209 Ars Magna 20 ASCII 66, 158, 212, 230 absolute prime 65, 146, 251 atomic number 44, 51, 64, 65 abundant number 103, 156 Australopithecus afarensis 46 aibohphobia 19 autism 85 aliquot sequence 13, 98 autobiographical prime 192 almost-all-even-digits prime 251 averaging sets 186 almost-equipandigital prime 251 alphabet code 50, 52, 61, 65, 73, Babbage 18, 146 81, 83 Babbage (portrait) 147 alphaprime code 83, 92, 110 balanced prime 12, 48, 113, 251 alternate-digit prime 251 Balog 104, 159 Amdahl Six 38 Balog cube 104 American Mathematical Society baseball 38, 97, 101, 116, 127, 70, 102, 196, 270 129 Antikythera mechanism 44 beast number 109, 129, 202, 204 apocalyptic number 72 beastly prime 142, 155, 229, 251 Apollonius 101 bemirp 113, 191, 210, 251 Archimedean solid 19 Bernoulli number 84, 94, 102 Archimedes 25, 33, 101, 167 Bernoulli triangle 214 { Page 287 { Bertrand prime Subject Index Bertrand prime 211 composite-digit prime 59, 136, Bertrand's postulate 111, 211, 252 252 computer mouse 187 Bible 23, 45, 49, 50, 59, 72, 83, congruence 252 85, 109, 158, 194, 216, 235, congruent prime 29, 196, 203, 236 213, 222, 227, -
A Proof of the Odd Perfect Number Conjecture
RFSC 04-01 Revised A PROOF OF THE ODD PERFECT NUMBER CONJECTURE Simon Davis Abstract. It is sufficient to prove that there is an excess of prime factors in the product of repunits with odd prime bases defined by the sum of divisors of the integer N = 4m+1 ℓ 2αi (4k +1) i=1 qi to establish that there do not exist any odd integers with equality between σ(N) and 2N. The existence of distinct prime divisors in the repunits in σ(N) Q follows from a theorem on the primitive divisors of the Lucas sequences U2αi+1(qi +1, qi) and U2αj +1(qj + 1, qj) with qi, qj, 2αi + 1, 2αj + 1 being odd primes. The occurrence 2α +1 4m+2 q i −1 of new prime divisors in each quotient (4k+1) −1 , i , i = 1,...,ℓ also implies 4k qi−1 that the square root of the product of 2(4k + 1) and the sequence of repunits will not be rational unless the primes are matched. Although a finite set of solutions to the rationality condition for the existence of odd perfect numbers is obtained, it is verified that they all σ(N) satisfy N 6= 2 because the repunits in the product representing σ(N) introduce new prime divisors. Minimization of the number of prime divisors in σ(N) leads to an infinite arXiv:hep-th/0401052v3 31 May 2008 set of repunits of increasing mangitude or prime equations with no integer solutions. It is proven then that there exist no odd perfect numbers. -
1. Introduction
Available online at http://scik.org J. Math. Comput. Sci. 2 (2012), No. 5, 1257-1268 ISSN: 1927-5307 CHROMATIC NUMBER OF GRAPHS WITH SPECIAL DISTANCE SETS-III V.YEGNANARAYANAN∗ AND A.PARTHIBAN Department of Mathematics, Velammal Engineering College, Chennai, India Abstract. An integer distance graph is a graph G(Z; D) with the set of integers as vertex set and an edge joining two vertices u and v if and only if j u − v j2 D where D is a subset of the positive integers. It is known that χ(G(Z; P )) = 4 where P is a set of prime numbers. In this paper we have identified the class three sets when the distance set D is not only a subset of primes P but also a special class of primes like Markov primes, Bell primes, Dihedral primes, Mills primes, Ramanujan primes, Quartan primes, Isolated primes and Thabit Number primes. We also indicate the existence of an alternative formulation for a prime distance graph and raise certain open problems. Keywords: Primes, Chromatic Number, Distance Graphs. 2000 AMS Subject Classification: 05C15 1. Introduction The graphs considered in this paper are simple and undirected. A k-coloring of a graph G is an assignment of k different colors to the vertices of G such that adjacent vertices receive different colors. The minimum cardinality k for which G has a k-coloring is called the chromatic number of G and is denoted by χ(G): ∗Corresponding author Received March 22, 2012 1257 1258 V.YEGNANARAYANAN∗ AND A.PARTHIBAN We begin with the plane coloring problem.