DOCSLIB.ORG

UIL NS and Mathematics Special Topics Session

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
UIL NS and Mathematics Special Topics Session UIL NS and Mathematics Special Topics Session Special session for Not the Ordinary Types of Numbers (Part 1) UIL NS and Mathematics Special Topics Session Larry White Box 25 Millersview, Tx 76862 325-483-5446 [email protected] UIL State NS & Mathematics Contest Director Be sure to ... Ø sign in; Ø clean up your area when you leave; Ø and “HAVE A GREAT DAY!” UIL NS and Mathematics Special Topics Session UIL Number Sense and Mathematics Constitution and Contest Rules (C&CR) ??? Any rules need discussing or clarified ??? UIL NS and Mathematics Special Topics Session No special topics this year Items of emphasis will include: Functions Circles Platonic Solids Patterns Fibonacci UIL NS and Mathematics Special Topics Session Two new mathematicians Franciscus Vieta Marin Mersenne UIL NS and Mathematics Special Topics Session Two new special types of numbers Germain Primes Mersenne Primes UIL NS and Mathematics Special Topics Session Not the Ordinary Types of Numbers (Part 1) Special Primes UIL NS and Mathematics Special Topics Session Germain Primes These are Germain Primes: 3, 11, 29, … These are not Germain Primes: 7, 13, 31, … UIL NS and Mathematics Special Topics Session Germain Primes If P is prime and 2P + 1 is prime, then P is a Germain prime. 2 ? 41 ? 37 ? UIL NS and Mathematics Special Topics Session Mersenne Primes These are Mersenne Primes: 2, 3, 5, … These are not Mersenne Primes: 11, 23, 31, … UIL NS and Mathematics Special Topics Session Mersenne Primes P If P is prime and 2 -1 is prime, then P is a Mersenne prime. 7 ? 13 ? 29 ? UIL NS and Mathematics Special Topics Session Mersenne Primes titanic primes --- Mersenne primes with more than 1000 digits gigantic primes --- Mersenne primes with more than 10,000 digits megaprimes --- Mersenne primes with more than 1,000,000 digits UIL NS and Mathematics Special Topics Session Mersenne Primes Note: The ten largest known Mersenne primes are the biggest prime numbers ever identified thus far. Note: The largest Mersenne primes is 2 57,885,161 – 1 It was discovered in 2013 and has more than 17,000,000 digits. UIL NS and Mathematics Special Topics Session Pythagorean Primes These are Pythagorean Primes: 5, 13, 41, … These are not Pythagorean Primes: 3, 11, 23, … UIL NS and Mathematics Special Topics Session Pythagorean Primes If N is a natural number and P = 4N + 1 is prime, then then P is a Pythagorean prime 2 ? 7 ? 31 ? UIL NS and Mathematics Special Topics Session Factorial Primes These are Factorial Primes: 3, 5, 7, … These are not Factorial Primes: 11, 13, 17, … UIL NS and Mathematics Special Topics Session Factorial Primes If N is a natural number and P = N! + 1 or P = N! - 1is prime, then then P is a Factorial prime. 2 ? 17 ? 23 ? UIL NS and Mathematics Special Topics Session Not the Ordinary Types of Numbers (Part 1) There are some real interesting things to learn about these special primes as well as other types of primes. Do some investigating and enjoy. HAVE A GREAT DAY! .
Load more

Recommended publications
  • A NOTE on the GAUSSIAN MOAT PROBLEM Real Axis
    A NOTE ON THE GAUSSIAN MOAT PROBLEM MADHUPARNA DAS Abstract. The Gaussian moat problem asks whether it is possible to find an infinite sequence of distinct Gaussian prime numbers such that the difference between consecutive numbers in the sequence is bounded. In this paper, we have proved that the answer is ‘No’, that is an infinite sequence of distinct Gaussian prime numbers can not be bounded by an absolute constant, for the Gaussian primes p = a2 + b2 with a, b =6 0. We consider each prime (a, b) as a lattice point on the complex plane and use their properties to prove the main result. 1. Introduction Primes are always interesting topics for mathematicians. After lots of research work still, we don’t have any explicit formula for the distribution of the natural prime numbers. Prime number theorem gives an asymptotic formula for primes with some bounded error. Also, the prime number theorem implies that there are arbitrarily large gaps in the sequence of prime numbers, and this can also be proved directly: for any n, the n 1 consecutive numbers n!+2,n!+3,...,n!+ n are all composite. It’s very easy for natural primes− that it is not possible to walk to infinity with stepping on the natural prime numbers with bounded length gap. Now we can think about the same problem for the complex prime numbers or explicitly for ‘Gaussian Primes’. So, the question says: “Whether it is possible to find an infinite sequence of distinct Gaussian prime numbers such that the difference between consecutive numbers in the sequence is bounded.
    [Show full text]
  • Miscellaneous HOL Examples
    Miscellaneous HOL Examples December 12, 2016 Contents 1 Some Isar command definitions 15 1.1 Diagnostic command: no state change............. 16 1.2 Old-style global theory declaration............... 16 1.3 Local theory specification.................... 16 2 Infinite Sets and Related Concepts 17 2.1 Infinitely Many and Almost All................. 18 2.2 Enumeration of an Infinite Set................. 21 3 Ad Hoc Overloading 24 3.1 Plain Ad Hoc Overloading.................... 24 3.2 Adhoc Overloading inside Locales................ 25 4 Permutation Types 27 5 Example of Declaring an Oracle 29 5.1 Oracle declaration........................ 29 5.2 Oracle as low-level rule...................... 29 5.3 Oracle as proof method..................... 30 6 Abstract Natural Numbers primitive recursion 34 7 Proof by guessing 37 8 Examples of function definitions 38 8.1 Very basic............................. 38 8.2 Currying.............................. 38 8.3 Nested recursion......................... 38 8.3.1 Here comes McCarthy's 91-function.......... 39 8.4 More general patterns...................... 39 8.4.1 Overlapping patterns................... 39 8.4.2 Guards.......................... 40 1 8.5 Mutual Recursion......................... 41 8.6 Definitions in local contexts................... 41 8.7 fun-cases ............................. 42 8.7.1 Predecessor........................ 42 8.7.2 List to option....................... 42 8.7.3 Boolean Functions.................... 43 8.7.4 Many parameters..................... 43 8.8 Partial Function Definitions................... 43 8.9 Regression tests.......................... 43 8.9.1 Context recursion.................... 44 8.9.2 A combination of context and nested recursion.... 44 8.9.3 Context, but no recursive call.............. 44 8.9.4 Tupled nested recursion................. 44 8.9.5 Let............................. 44 8.9.6 Abbreviations......................
    [Show full text]
  • On Prime Numbers and Related Applications
    International Journal of Innovative Technology and Exploring Engineering (IJITEE) ISSN: 2278-3075, Volume-8 Issue-12, October 2019 On Prime Numbers and Related Applications V Yegnanarayanan, Veena Narayanan, R Srikanth Abstract: In this paper we probed some interesting aspects of Our motivating factor for this work stems from the classic primorial and factorial primes. We did some numerical analysis result of called Prime Number Theorem which beautifully about the distribution of prime numbers and tabulated our models the statistical behavioral pattern of huge primes viz., findings. Also, we pointed out certain interesting facts about the utility value of the study of prime numbers and their distributions chance for an arbitrarily picked 푛휖푁 to be a prime number is in control engineering and Brain networks. inverse in proportion to log(푛). Keywords: Factorial primes, Primorial primes, Prime numbers. II. RESULTS AND DISCUSSIONS I. INTRODUCTION A. Discussion on Primorial and Factorial Primes The irregular distribution of prime numbers makes it Our journey is driven by fascinating characteristics of difficult for us to presume and discover. Euclid about 2300 primordial primes and factorial primes. A primordial years before established that primes are infinitely many. Till function denoted 푃# is the product of all primes till P and date we are not aware of a formula for finding the nth 푛 includes P. The factorial function is 푛! = 푖=1 푖. A natural prime. It appears as if they are randomly embedded in N and curiosity pushes us to probe large primes. In order to do this, may be because of this that sequences of primes do not tend we need to get an idea by analyzing numbers such as to emanate from naturally happening processes.
    [Show full text]
  • Metallic Ratios in Primitive Pythagorean Triples : Metallic Means Embedded in Pythagorean Triangles and Other Right Triangles
    Journal of Advances in Mathematics Vol 20 (2021) ISSN: 2347-1921 https://rajpub.com/index.php/jam DOI: https://doi.org/10.24297/jam.v20i.9088 Metallic Ratios in Primitive Pythagorean Triples : Metallic Means embedded in Pythagorean Triangles and other Right Triangles Dr. Chetansing Rajput M.B.B.S. Nair Hospital (Mumbai University) India, Asst. Commissioner (Govt. of Maharashtra) Email: [email protected] Website: https://goldenratiorajput.com/ Lecture Link 1 : https://youtu.be/LFW1saNOp20 Lecture Link 2 : https://youtu.be/vBfVDaFnA2k Lecture Link 3 : https://youtu.be/raosniXwRhw Lecture Link 4 : https://youtu.be/74uF4sBqYjs Lecture Link 5 : https://youtu.be/Qh2B1tMl8Bk Abstract The Primitive Pythagorean Triples are found to be the purest expressions of various Metallic Ratios. Each Metallic Mean is epitomized by one particular Pythagorean Triangle. Also, the Right Angled Triangles are found to be more “Metallic” than the Pentagons, Octagons or any other (n2+4)gons. The Primitive Pythagorean Triples, not the regular polygons, are the prototypical forms of all Metallic Means. Keywords: Metallic Mean, Pythagoras Theorem, Right Triangle, Metallic Ratio Triads, Pythagorean Triples, Golden Ratio, Pascal’s Triangle, Pythagorean Triangles, Metallic Ratio A Primitive Pythagorean Triple for each Metallic Mean (훅n) : Author’s previous paper titled “Golden Ratio” cited by the Wikipedia page on “Metallic Mean” [1] & [2], among other works mentioned in the References, have already highlighted the underlying proposition that the Metallic Means
    [Show full text]
  • Integer Sequences
    UHX6PF65ITVK Book > Integer sequences Integer sequences Filesize: 5.04 MB Reviews A very wonderful book with lucid and perfect answers. It is probably the most incredible book i have study. Its been designed in an exceptionally simple way and is particularly just after i finished reading through this publication by which in fact transformed me, alter the way in my opinion. (Macey Schneider) DISCLAIMER | DMCA 4VUBA9SJ1UP6 PDF > Integer sequences INTEGER SEQUENCES Reference Series Books LLC Dez 2011, 2011. Taschenbuch. Book Condition: Neu. 247x192x7 mm. This item is printed on demand - Print on Demand Neuware - Source: Wikipedia. Pages: 141. Chapters: Prime number, Factorial, Binomial coeicient, Perfect number, Carmichael number, Integer sequence, Mersenne prime, Bernoulli number, Euler numbers, Fermat number, Square-free integer, Amicable number, Stirling number, Partition, Lah number, Super-Poulet number, Arithmetic progression, Derangement, Composite number, On-Line Encyclopedia of Integer Sequences, Catalan number, Pell number, Power of two, Sylvester's sequence, Regular number, Polite number, Ménage problem, Greedy algorithm for Egyptian fractions, Practical number, Bell number, Dedekind number, Hofstadter sequence, Beatty sequence, Hyperperfect number, Elliptic divisibility sequence, Powerful number, Znám's problem, Eulerian number, Singly and doubly even, Highly composite number, Strict weak ordering, Calkin Wilf tree, Lucas sequence, Padovan sequence, Triangular number, Squared triangular number, Figurate number, Cube, Square triangular
    [Show full text]
  • New Congruences and Finite Difference Equations For
    New Congruences and Finite Difference Equations for Generalized Factorial Functions Maxie D. Schmidt University of Washington Department of Mathematics Padelford Hall Seattle, WA 98195 USA [email protected] Abstract th We use the rationality of the generalized h convergent functions, Convh(α, R; z), to the infinite J-fraction expansions enumerating the generalized factorial product se- quences, pn(α, R)= R(R + α) · · · (R + (n − 1)α), defined in the references to construct new congruences and h-order finite difference equations for generalized factorial func- tions modulo hαt for any primes or odd integers h ≥ 2 and integers 0 ≤ t ≤ h. Special cases of the results we consider within the article include applications to new congru- ences and exact formulas for the α-factorial functions, n!(α). Applications of the new results we consider within the article include new finite sums for the α-factorial func- tions, restatements of classical necessary and sufficient conditions of the primality of special integer subsequences and tuples, and new finite sums for the single and double factorial functions modulo integers h ≥ 2. 1 Notation and other conventions in the article 1.1 Notation and special sequences arXiv:1701.04741v1 [math.CO] 17 Jan 2017 Most of the conventions in the article are consistent with the notation employed within the Concrete Mathematics reference, and the conventions defined in the introduction to the first articles [11, 12]. These conventions include the following particular notational variants: ◮ Extraction of formal power series coefficients. The special notation for formal n k power series coefficient extraction, [z ] k fkz :7→ fn; ◮ Iverson’s convention.
    [Show full text]
  • Searching for a Counterexample to Kurepa's Conjecture
    SEARCHING FOR A COUNTEREXAMPLE TO KUREPA'S CONJECTURE VLADICA ANDREJIC´ AND MILOS TATAREVIC Abstract. Kurepa's conjecture states that there is no odd prime p that di- vides !p = 0!+1!+···+(p−1)!. We search for a counterexample to this conjec- ture for all p < 234. We introduce new optimization techniques and perform the computation using graphics processing units. Additionally, we consider the generalized Kurepa's left factorial given by !kn = (0!)k +(1!)k +···+((n−1)!)k, and show that for all integers 1 < k < 100 there exists an odd prime p such that p j!kp. 1. Introduction ¯Duro Kurepa defines an arithmetic function !n, known as the left factorial, by !n = 0! + 1! + ··· + (n − 1)! [14]. He conjectures that GCD(!n; n!) = 2 holds for all integers n > 1. This conjecture, originally introduced in 1971 [14], is also listed in [10, Section B44] and remains an open problem. For additional information, the reader can consult the expository article [12]. It is easy to see that the statement above is equivalent to the statement that !n is not divisible by n > 2, which can be reduced to primes. Thus, Kurepa's conjecture states that there is no odd prime p such that p divides !p. There have been numerous attempts to solve this problem, mostly by searching for a counterexample. The results of these attempts are listed in the accompanying table. Upper Bound Author Year p < 3 · 105 Z.ˇ Mijajlovi´c[16] 1990 p < 106 G. Gogi´c[9] 1991 p < 3 · 106 B. Maleˇsevi´c[15] 1998 p < 223 M.
    [Show full text]
  • Factoring of Large Numbers
    FACTORING OF LARGE NUMBERS It is well known that any number N can be represented as the product of a finite number of primes taken to the specified integer powers. Thus- 13867 = (7)2 (283) with sqrt(13867)= 117.756.. 23974723= (151) (179) (887) with sqrt(23974723)= 4896.398.. 43567217=(5021) (8677) with sqrt(43567217)= 6600.546.. 67537=67537 The first three examples represent composite numbers while the last is a prime. Now we know from some of our earlier notes that all primes above p=3 have the form- p=6n1 , where n=1,2,3,4,… This is a necessary but not sufficient condition for primes since there can be many composite numbers which also have this same form. We want to show here how to distinguish between a composite and prime number satisfying the 6n1 condition and from this deduce the product form of any number N. Since we don’t know before hand into how many prime terms a number N will factor, we begin with the safe bet that if N is a semi-prime N=pq, then p<sqrt(N), provided that p<q. If it is composed of three terms N=pqr, even when some of the factors appear multiple times, then it is save to assume that at least one of the primes is less than N1/3 and so on. So we definitely can state, regardless of the number of prime products present in N, at least one of them will always be less than the square-root of N unless the number itself is already a prime.
    [Show full text]
  • Numbers 1 to 100
    Numbers 1 to 100 PDF generated using the open source mwlib toolkit. See http://code.pediapress.com/ for more information. PDF generated at: Tue, 30 Nov 2010 02:36:24 UTC Contents Articles −1 (number) 1 0 (number) 3 1 (number) 12 2 (number) 17 3 (number) 23 4 (number) 32 5 (number) 42 6 (number) 50 7 (number) 58 8 (number) 73 9 (number) 77 10 (number) 82 11 (number) 88 12 (number) 94 13 (number) 102 14 (number) 107 15 (number) 111 16 (number) 114 17 (number) 118 18 (number) 124 19 (number) 127 20 (number) 132 21 (number) 136 22 (number) 140 23 (number) 144 24 (number) 148 25 (number) 152 26 (number) 155 27 (number) 158 28 (number) 162 29 (number) 165 30 (number) 168 31 (number) 172 32 (number) 175 33 (number) 179 34 (number) 182 35 (number) 185 36 (number) 188 37 (number) 191 38 (number) 193 39 (number) 196 40 (number) 199 41 (number) 204 42 (number) 207 43 (number) 214 44 (number) 217 45 (number) 220 46 (number) 222 47 (number) 225 48 (number) 229 49 (number) 232 50 (number) 235 51 (number) 238 52 (number) 241 53 (number) 243 54 (number) 246 55 (number) 248 56 (number) 251 57 (number) 255 58 (number) 258 59 (number) 260 60 (number) 263 61 (number) 267 62 (number) 270 63 (number) 272 64 (number) 274 66 (number) 277 67 (number) 280 68 (number) 282 69 (number) 284 70 (number) 286 71 (number) 289 72 (number) 292 73 (number) 296 74 (number) 298 75 (number) 301 77 (number) 302 78 (number) 305 79 (number) 307 80 (number) 309 81 (number) 311 82 (number) 313 83 (number) 315 84 (number) 318 85 (number) 320 86 (number) 323 87 (number) 326 88 (number)
    [Show full text]
  • Vedic Math World MATH, LOGIC and MUCH MORE…
    March2VEDIC021 MATH WORLD VolumeMarch 2, 202 Issue1 3 A unique initiative of Vedic Math Learning System (VMLS) Volume 2, Issue 3 A unique initiative of Vedic Math Learning System (VMLS) INDEX Vedic Math World MATH, LOGIC AND MUCH MORE… THE WORLD CELEBRATED 흅- DAY AS INTERNATIONAL DAY OF MATHEMATICS MADHAVA ARYABHATA World’s first person Gave the value 휋 = 3.1416 to estimate the in 5th century exact value of 휋 (pi) BHASKARACHARYA Defined the value of pi as 22/7 www.vedicmathworld.com/magazine PAGE 1 OF 40 VEDIC MATH WORLD March2021 A unique initiative of Vedic Math Learning System (VMLS) Volume 2, Issue 3 INSIDE THIS ISSUE FOCUS March SHOW-CASING THE MAJOR ATTRACTIONS & INDEX 2021 (SMART-LINKS) Page#3 Page#22 IMPORTANT LINKS ‘GANITANAND’ INDEX A summary of all links available on one page Magic of ‘pi’ ! Page#4 Page#23 A SPECIAL MESSAGE RESPECT YOUR ROOTS! Hon’ble Union Minister’s words History of an interesting mathematical number ‘’ (800BCE – 1200AD) Page#5 A SPECIAL MESSAGE Page#27 Hon’ble Education Minister’s words DID YOU KNOW? Page #6 Practical applications of ‘pi’ in real world SANSTHAPAKS’ NOTE Page #29 The concept of ‘pi’ was very first given during 800BCE in MATHOPINION India! Mathamagic of March Page#7 Page #32 MATH NEWS N’ UPDATES THEMATIC SPECIALS About math around the world ‘Pi’ Day celebrated worldwide as ‘International Mathematics Day’ Page#10 POWER OF ‘VEDIC MATH’ Page #34 निनिलम् िवतश्चरमं दशतः MATH-MANTRA QUIZ An important Vedic Math Sutra: QUIZ Based on this Magazine Content Page #13 Math -Mantra QUIZ#8 LOGIC QUEST DIRECT
    [Show full text]
  • About the Author and This Text
    1 PHUN WITH DIK AND JAYN BREAKING RSA CODES FOR FUN AND PROFIT A supplement to the text CALCULATE PRIMES By James M. McCanney, M.S. RSA-2048 has 617 decimal digits (2,048 bits). It is the largest of the RSA numbers and carried the largest cash prize for its factorization, US$200,000. The largest factored RSA number is 768 bits long (232 decimal digits). According to the RSA Corporation the RSA-2048 may not be factorable for many years to come, unless considerable advances are made in integer factorization or computational power in the near future Your assignment after reading this book (using the Generator Function of the book Calculate Primes) is to find the two prime factors of RSA-2048 (listed below) and mail it to the RSA Corporation explaining that you used the work of Professor James McCanney. RSA-2048 = 25195908475657893494027183240048398571429282126204032027777 13783604366202070759555626401852588078440691829064124951508 21892985591491761845028084891200728449926873928072877767359 71418347270261896375014971824691165077613379859095700097330 45974880842840179742910064245869181719511874612151517265463 22822168699875491824224336372590851418654620435767984233871 84774447920739934236584823824281198163815010674810451660377 30605620161967625613384414360383390441495263443219011465754 44541784240209246165157233507787077498171257724679629263863 56373289912154831438167899885040445364023527381951378636564 391212010397122822120720357 jmccanneyscience.com press - ISBN 978-0-9828520-3-3 $13.95US NOT FOR RESALE Copyrights 2006, 2007, 2011, 2014
    [Show full text]
  • Logarithm of the Exponents in the Prime Factorization of the Factorial
    International Mathematical Forum, Vol. 12, 2017, no. 13, 643 - 649 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/imf.2017.7543 Logarithm of the Exponents in the Prime Factorization of the Factorial Rafael Jakimczuk Divisi´onMatem´atica,Universidad Nacional de Luj´an Buenos Aires, Argentina Copyright c 2017 Rafael Jakimczuk. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduc- tion in any medium, provided the original work is properly cited. Abstract P In this note we study the sum 2≤p≤n log E(p) where E(p) is the exponent of the prime p in the prime factorization of n!, the sum P 2≤p≤n HE(p), where Hk denotes the k-th harmonic number and an- other sums. We also consider the generalized harmonic numbers Hk;m of order m and obtain a strong connection between these sums and the Riemann zeta function. Mathematics Subject Classification: 11A99, 11B99 Keywords: Factorial, prime factorization, Exponents, Logarithm 1 Introduction Let us consider the prime factorization of a positive integer a s1 s2 sk a = p1 p2 ··· pk ; where p1; p2; : : : ; pk are the different primes in the prime factorization and s1; s2; : : : ; sk are the exponents. The total number of prime factors in the prime factorization is denoted Ω(a) (see [1, chapter XXII]), that is, Ω(a) = s1 + s2 + ··· + sk: 644 Rafael Jakimczuk Let us consider the factorial n!, the exponent of a prime p in its prime factor- ization will be denoted E(p). Then we can write the prime factorization of n! in the form n! = Y pE(p) 2≤p≤n since, clearly, the primes that appear in the prime factorization of n! are the primes not exceeding n.
    [Show full text]