Understanding Exponents Name: Rewrite the Number As a Multiplication Problem

Total Page:16

File Type:pdf, Size:1020Kb

Understanding Exponents Name: Rewrite the Number As a Multiplication Problem Understanding Exponents Name: Rewrite the number as a multiplication problem. Answers 1) 54 2 2) 3 1. 5×5×5×5 3) 25 2. 3×3 4) 53 5) 5 7 3. 2×2×2×2×2 4. 5×5×5 Rewrite the multiplication problem in exponential notation. 5. 7×7×7×7×7 6) 7×7×7×7×7 5 7) 9×9×9×9 6. 7 8) 3×3×3 4 7. 9 9) 4×4×4 3 10) 5×5×5×5×5 8. 3 3 9. 4 5 10. 5 11. 81 Solve the problems. 11) 92 12. 64 12) 43 2 13) 3 13. 9 14) 23 15) 52 14. 8 16) What is 2 to the power of two? 15. 25 17) What is 5 cubed? 18) What is 9 to the power of three? 16. 4 19) What is 6 to the power of three? 17. 125 20) What is 7 to the power of two? 18. 729 19. 216 20. 49 1-10 95 90 85 80 75 70 65 60 55 50 Math www.CommonCoreSheets.com 1 11-20 45 40 35 30 25 20 15 10 5 0 Understanding Exponents Name: Answer Key Rewrite the number as a multiplication problem. Answers 1) 54 2 2) 3 1. 5×5×5×5 3) 25 2. 3×3 4) 53 5) 5 7 3. 2×2×2×2×2 4. 5×5×5 Rewrite the multiplication problem in exponential notation. 5. 7×7×7×7×7 6) 7×7×7×7×7 5 7) 9×9×9×9 6. 7 8) 3×3×3 4 7. 9 9) 4×4×4 3 10) 5×5×5×5×5 8. 3 3 9. 4 5 10. 5 11. 81 Solve the problems. 11) 92 12. 64 12) 43 2 13) 3 13. 9 14) 23 15) 52 14. 8 16) What is 2 to the power of two? 15. 25 17) What is 5 cubed? 18) What is 9 to the power of three? 16. 4 19) What is 6 to the power of three? 17. 125 20) What is 7 to the power of two? 18. 729 19. 216 20. 49 1-10 95 90 85 80 75 70 65 60 55 50 Math www.CommonCoreSheets.com 1 11-20 45 40 35 30 25 20 15 10 5 0 Understanding Exponents Name: Rewrite the number as a multiplication problem. Answers 1) 22 4 2) 7 1. 2×2 3) 85 2. 7×7×7×7 4) 35 5) 3 2 3. 8×8×8×8×8 4. 3×3×3×3×3 Rewrite the multiplication problem in exponential notation. 5. 2×2×2 6) 4×4×4×4 4 7) 4×4×4 6. 4 8) 9×9×9 3 7. 4 9) 4×4 3 10) 9×9×9×9×9 8. 9 2 9. 4 5 10. 9 11. 16 Solve the problems. 11) 42 12. 25 12) 52 2 13) 3 13. 9 14) 92 15) 93 14. 81 16) What is 8 to the power of two? 15. 729 17) What is 8 to the power of three? 18) What is 7 squared? 16. 64 19) What is 6 to the power of two? 17. 512 20) What is 3 cubed? 18. 49 19. 36 20. 27 1-10 95 90 85 80 75 70 65 60 55 50 Math www.CommonCoreSheets.com 2 11-20 45 40 35 30 25 20 15 10 5 0 Understanding Exponents Name: Answer Key Rewrite the number as a multiplication problem. Answers 1) 22 4 2) 7 1. 2×2 3) 85 2. 7×7×7×7 4) 35 5) 3 2 3. 8×8×8×8×8 4. 3×3×3×3×3 Rewrite the multiplication problem in exponential notation. 5. 2×2×2 6) 4×4×4×4 4 7) 4×4×4 6. 4 8) 9×9×9 3 7. 4 9) 4×4 3 10) 9×9×9×9×9 8. 9 2 9. 4 5 10. 9 11. 16 Solve the problems. 11) 42 12. 25 12) 52 2 13) 3 13. 9 14) 92 15) 93 14. 81 16) What is 8 to the power of two? 15. 729 17) What is 8 to the power of three? 18) What is 7 squared? 16. 64 19) What is 6 to the power of two? 17. 512 20) What is 3 cubed? 18. 49 19. 36 20. 27 1-10 95 90 85 80 75 70 65 60 55 50 Math www.CommonCoreSheets.com 2 11-20 45 40 35 30 25 20 15 10 5 0 Understanding Exponents Name: Rewrite the number as a multiplication problem. Answers 1) 95 5 2) 4 1. 9×9×9×9×9 3) 64 2. 4×4×4×4×4 4) 62 5) 3 4 3. 6×6×6×6 4. 6×6 Rewrite the multiplication problem in exponential notation. 5. 4×4×4 6) 4×4×4×4 4 7) 3×3 6. 4 8) 5×5×5×5×5 2 7. 3 9) 3×3×3 5 10) 4×4×4×4×4 8. 5 3 9. 3 5 10. 4 11. 216 Solve the problems. 11) 63 12. 81 12) 92 3 13) 8 13. 512 14) 22 15) 32 14. 4 16) What is 9 cubed? 15. 9 17) What is 7 to the power of two? 18) What is 8 squared? 16. 729 19) What is 7 to the power of three? 17. 49 20) What is 5 cubed? 18. 64 19. 343 20. 125 1-10 95 90 85 80 75 70 65 60 55 50 Math www.CommonCoreSheets.com 3 11-20 45 40 35 30 25 20 15 10 5 0 Understanding Exponents Name: Answer Key Rewrite the number as a multiplication problem. Answers 1) 95 5 2) 4 1. 9×9×9×9×9 3) 64 2. 4×4×4×4×4 4) 62 5) 3 4 3. 6×6×6×6 4. 6×6 Rewrite the multiplication problem in exponential notation. 5. 4×4×4 6) 4×4×4×4 4 7) 3×3 6. 4 8) 5×5×5×5×5 2 7. 3 9) 3×3×3 5 10) 4×4×4×4×4 8. 5 3 9. 3 5 10. 4 11. 216 Solve the problems. 11) 63 12. 81 12) 92 3 13) 8 13. 512 14) 22 15) 32 14. 4 16) What is 9 cubed? 15. 9 17) What is 7 to the power of two? 18) What is 8 squared? 16. 729 19) What is 7 to the power of three? 17. 49 20) What is 5 cubed? 18. 64 19. 343 20. 125 1-10 95 90 85 80 75 70 65 60 55 50 Math www.CommonCoreSheets.com 3 11-20 45 40 35 30 25 20 15 10 5 0 Understanding Exponents Name: Rewrite the number as a multiplication problem. Answers 1) 95 3 2) 3 1. 9×9×9×9×9 3) 84 2. 3×3×3 4) 93 5) 3 6 3. 8×8×8×8 4. 9×9×9 Rewrite the multiplication problem in exponential notation. 5. 6×6×6 6) 9×9 2 7) 6×6×6×6 6. 9 8) 2×2 4 7. 6 9) 6×6 2 10) 3×3 8. 2 2 9. 6 2 10. 3 11. 27 Solve the problems. 11) 33 12. 9 12) 32 3 13) 7 13. 343 14) 23 15) 43 14. 8 16) What is 6 to the power of three? 15. 64 17) What is 8 to the power of two? 18) What is 5 to the power of three? 16. 216 19) What is 2 to the power of two? 17. 64 20) What is 6 to the power of two? 18. 125 19. 4 20. 36 1-10 95 90 85 80 75 70 65 60 55 50 Math www.CommonCoreSheets.com 4 11-20 45 40 35 30 25 20 15 10 5 0 Understanding Exponents Name: Answer Key Rewrite the number as a multiplication problem. Answers 1) 95 3 2) 3 1. 9×9×9×9×9 3) 84 2. 3×3×3 4) 93 5) 3 6 3. 8×8×8×8 4. 9×9×9 Rewrite the multiplication problem in exponential notation. 5. 6×6×6 6) 9×9 2 7) 6×6×6×6 6. 9 8) 2×2 4 7. 6 9) 6×6 2 10) 3×3 8. 2 2 9. 6 2 10. 3 11. 27 Solve the problems. 11) 33 12. 9 12) 32 3 13) 7 13. 343 14) 23 15) 43 14. 8 16) What is 6 to the power of three? 15. 64 17) What is 8 to the power of two? 18) What is 5 to the power of three? 16. 216 19) What is 2 to the power of two? 17. 64 20) What is 6 to the power of two? 18. 125 19. 4 20. 36 1-10 95 90 85 80 75 70 65 60 55 50 Math www.CommonCoreSheets.com 4 11-20 45 40 35 30 25 20 15 10 5 0 Understanding Exponents Name: Rewrite the number as a multiplication problem. Answers 1) 55 2 2) 8 1. 5×5×5×5×5 3) 73 2. 8×8 4) 45 5) 3 4 3. 7×7×7 4. 4×4×4×4×4 Rewrite the multiplication problem in exponential notation. 5.
Recommended publications
  • Microfilmed 1996 Information to Users
    UMI MICROFILMED 1996 INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleed through, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand comer and continuing from left to right in equal sections with small overlaps. Each original is also photographed in one exposure and is included in reduced form at the back of the book. Photographs included in the original manuscript have been reproduced xerographically in this copy. Higher quality 6" x 9" black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order. UMI A Bell & Howell Information Company 300 NonhZeeb Road, Ann Arbor MI 48106-1346 USA 313/761-4700 800/521-0600 Interpoint Distance Methods for the Analysis of High Dimensional Data DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in the Graduate School of The Ohio State University By John Lawrence, B.
    [Show full text]
  • On Terms of Generalized Fibonacci Sequences Which Are Powers of Their Indexes
    mathematics Article On Terms of Generalized Fibonacci Sequences which are Powers of their Indexes Pavel Trojovský Department of Mathematics, Faculty of Science, University of Hradec Králové, 500 03 Hradec Králové, Czech Republic, [email protected]; Tel.: +42-049-333-2860 Received: 29 June 2019; Accepted: 31 July 2019; Published: 3 August 2019 (k) Abstract: The k-generalized Fibonacci sequence (Fn )n (sometimes also called k-bonacci or k-step Fibonacci sequence), with k ≥ 2, is defined by the values 0, 0, ... , 0, 1 of starting k its terms and such way that each term afterwards is the sum of the k preceding terms. This paper is devoted to the proof of the fact that the (k) t (2) 2 (3) 2 Diophantine equation Fm = m , with t > 1 and m > k + 1, has only solutions F12 = 12 and F9 = 9 . Keywords: k-generalized Fibonacci sequence; Diophantine equation; linear form in logarithms; continued fraction MSC: Primary 11J86; Secondary 11B39. 1. Introduction The well-known Fibonacci sequence (Fn)n≥0 is given by the following recurrence of the second order Fn+2 = Fn+1 + Fn, for n ≥ 0, with the initial terms F0 = 0 and F1 = 1. Fibonacci numbers have a lot of very interesting properties (see e.g., book of Koshy [1]). One of the famous classical problems, which has attracted a attention of many mathematicians during the last thirty years of the twenty century, was the problem of finding perfect powers in the sequence of Fibonacci numbers. Finally in 2006 Bugeaud et al. [2] (Theorem 1), confirmed these expectations, as they showed that 0, 1, 8 and 144 are the only perfect powers in the sequence of Fibonacci numbers.
    [Show full text]
  • Number Systems and Radix Conversion
    Number Systems and Radix Conversion Sanjay Rajopadhye, Colorado State University 1 Introduction These notes for CS 270 describe polynomial number systems. The material is not in the textbook, but will be required for PA1. We humans are comfortable with numbers in the decimal system where each po- sition has a weight which is a power of 10: units have a weight of 1 (100), ten’s have 10 (101), etc. Even the fractional apart after the decimal point have a weight that is a (negative) power of ten. So the number 143.25 has a value that is 1 ∗ 102 + 4 ∗ 101 + 3 ∗ 0 −1 −2 2 5 10 + 2 ∗ 10 + 5 ∗ 10 , i.e., 100 + 40 + 3 + 10 + 100 . You can think of this as a polyno- mial with coefficients 1, 4, 3, 2, and 5 (i.e., the polynomial 1x2 + 4x + 3 + 2x−1 + 5x−2+ evaluated at x = 10. There is nothing special about 10 (just that humans evolved with ten fingers). We can use any radix, r, and write a number system with digits that range from 0 to r − 1. If our radix is larger than 10, we will need to invent “new digits”. We will use the letters of the alphabet: the digit A represents 10, B is 11, K is 20, etc. 2 What is the value of a radix-r number? Mathematically, a sequence of digits (the dot to the right of d0 is called the radix point rather than the decimal point), : : : d2d1d0:d−1d−2 ::: represents a number x defined as follows X i x = dir i 2 1 0 −1 −2 = ::: + d2r + d1r + d0r + d−1r + d−2r + ::: 1 Example What is 3 in radix 3? Answer: 0.1.
    [Show full text]
  • Cheat Sheet of SSE/AVX Intrinsics, for Doing Arithmetic Ll F(Int Ind, Int K) { Return Dp[Ind][K]; } on Several Numbers at Once
    University of Bergen Garbage Collectors Davide Pallotti, Jan Soukup, Olav Røthe Bakken NWERC 2017 Nov 8, 2017 UiB template .bashrc .vimrc troubleshoot 1 tan v + tan w Contest (1) Any possible infinite recursion? tan(v + w) = Invalidated pointers or iterators? 1 − tan v tan w template.cpp Are you using too much memory? v + w v − w 15 lines Debug with resubmits (e.g. remapped signals, see Various). sin v + sin w = 2 sin cos #include <bits/stdc++.h> 2 2 using namespace std; Time limit exceeded: v + w v − w Do you have any possible infinite loops? cos v + cos w = 2 cos cos #define rep(i, a, b) for(int i = a; i < (b); ++i) What is the complexity of your algorithm? 2 2 #define trav(a, x) for(auto& a : x) Are you copying a lot of unnecessary data? (References) #define all(x) x.begin(), x.end() How big is the input and output? (consider scanf) (V + W ) tan(v − w)=2 = (V − W ) tan(v + w)=2 #define sz(x) (int)(x).size() Avoid vector, map. (use arrays/unordered_map) typedef long long ll; What do your team mates think about your algorithm? where V; W are lengths of sides opposite angles v; w. typedef pair<int, int> pii; typedef vector<int> vi; Memory limit exceeded: a cos x + b sin x = r cos(x − φ) What is the max amount of memory your algorithm should need? int main() { Are you clearing all datastructures between test cases? a sin x + b cos x = r sin(x + φ) cin.sync_with_stdio(0); cin.tie(0); cin.exceptions(cin.failbit); p 2 2 } Mathematics (2) where r = a + b ; φ = atan2(b; a).
    [Show full text]
  • Composite Numbers That Give Valid RSA Key Pairs for Any Coprime P
    information Article Composite Numbers That Give Valid RSA Key Pairs for Any Coprime p Barry Fagin ID Department of Computer Science, US Air Force Academy, Colorado Springs, CO 80840, USA; [email protected]; Tel.: +1-719-339-4514 Received: 13 August 2018; Accepted: 25 August 2018; Published: 28 August 2018 Abstract: RSA key pairs are normally generated from two large primes p and q. We consider what happens if they are generated from two integers s and r, where r is prime, but unbeknownst to the user, s is not. Under most circumstances, the correctness of encryption and decryption depends on the choice of the public and private exponents e and d. In some cases, specific (s, r) pairs can be found for which encryption and decryption will be correct for any (e, d) exponent pair. Certain s exist, however, for which encryption and decryption are correct for any odd prime r - s. We give necessary and sufficient conditions for s with this property. Keywords: cryptography; abstract algebra; RSA; computer science education; cryptography education MSC: [2010] 11Axx 11T71 1. Notation and Background Consider the RSA public-key cryptosystem and its operations of encryption and decryption [1]. Let (p, q) be primes, n = p ∗ q, f(n) = (p − 1)(q − 1) denote Euler’s totient function and (e, d) the ∗ Z encryption/decryption exponent pair chosen such that ed ≡ 1. Let n = Un be the group of units f(n) Z mod n, and let a 2 Un. Encryption and decryption operations are given by: (ae)d ≡ (aed) ≡ (a1) ≡ a mod n We consider the case of RSA encryption and decryption where at least one of (p, q) is a composite number s.
    [Show full text]
  • On Moduli for Which the Fibonacci Sequence Contains a Complete System of Residues S
    ON MODULI FOR WHICH THE FIBONACCI SEQUENCE CONTAINS A COMPLETE SYSTEM OF RESIDUES S. A. BURR Belt Telephone Laboratories, Inc., Whippany, New Jersey Shah [1] and Bruckner [2] have considered the problem of determining which moduli m have the property that the Fibonacci sequence {u }, de- fined in the usual way9 contains a complete system of residues modulo m, Following Shah we say that m is defective if m does not have this property, The results proved in [1] includes (I) If m is defective, so is any multiple of m; in particular, 8n is always defective. (II) if p is a prime not 2 or 5, p is defective unless p = 3 or 7 (mod 20). (in) If p is a prime = 3 or 7 (mod 20) and is not defective^ thenthe set {0, ±1, =tu3, ±u49 ±u5, • • • , ±u }9 where h = (p + l)/2 ? is a complete system of residues modulo p„ In [2], Bruckner settles the case of prime moduli by showing that all primes are defective except 2, 3S 5, and 7. In this paper we complete the work of Shah and Bruckner by proving the following re suit f which completely characterizes all defective and nondefective moduli. Theorem. A number m is not defective if and only if m has one of the following forms: k k k 5 , 2 * 5 f 4»5 , 3 k k 3 *5 5 6«5 , k k 7-5 , 14* 5* , where k ^ 0S j — 1. Thus almost all numbers are defective. We will prove a series of lem- mas, from which the theorem will follow directly, We first make some use- ful definitions.
    [Show full text]
  • Grade 7 Mathematics Strand 6 Pattern and Algebra
    GR 7 MATHEMATICS S6 1 TITLE PAGE GRADE 7 MATHEMATICS STRAND 6 PATTERN AND ALGEBRA SUB-STRAND 1: NUMBER PATTERNS SUB-STRAND 2: DIRECTED NUMBERS SUB-STRAND 3: INDICES SUB-STARND 4: ALGEBRA GR 7 MATHEMATICS S6 2 ACKNOWLEDGEMENT Acknowledgements We acknowledge the contributions of all Secondary and Upper Primary Teachers who in one way or another helped to develop this Course. Special thanks to the Staff of the mathematics Department of FODE who played active role in coordinating writing workshops, outsourcing lesson writing and editing processes, involving selected teachers of Madang, Central Province and NCD. We also acknowledge the professional guidance provided by the Curriculum Development and Assessment Division throughout the processes of writing and, the services given by the members of the Mathematics Review and Academic Committees. The development of this book was co-funded by GoPNG and World Bank. MR. DEMAS TONGOGO Principal- FODE . Written by: Luzviminda B. Fernandez SCO-Mathematics Department Flexible Open and Distance Education Papua New Guinea Published in 2016 @ Copyright 2016, Department of Education Papua New Guinea All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means electronic, mechanical, photocopying, recording or any other form of reproduction by any process is allowed without the prior permission of the publisher. ISBN: 978 - 9980 - 87 - 250 - 0 National Library Services of Papua New Guinea Printed by the Flexible, Open and Distance Education GR 7 MATHEMATICS S6 3 CONTENTS CONTENTS Page Secretary‟s Message…………………………………….…………………………………......... 4 Strand Introduction…………………………………….…………………………………………. 5 Study Guide………………………………………………….……………………………………. 6 SUB-STRAND 1: NUMBER PATTERNS ……………...….……….……………..………..
    [Show full text]
  • Estimating Indirect Parental Genetic Effects on Offspring Phenotypes Using Virtual Parental Genotypes Derived from Sibling and Half Sibling Pairs
    bioRxiv preprint doi: https://doi.org/10.1101/2020.02.21.959114; this version posted February 25, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY 4.0 International license. Estimating indirect parental genetic effects on offspring phenotypes using virtual parental genotypes derived from sibling and half sibling pairs Liang-Dar Hwang#1, Justin D Tubbs#2, Justin Luong1, Gunn-Helen Moen1,3, Pak C Sham*2,5,6, Gabriel Cuellar-Partida*1, David M Evans*1,7 1The University of Queensland Diamantina Institute, The University of Queensland, Brisbane, Australia 2Department of Psychiatry, The University of Hong Kong, Hong Kong SAR, China 3Institute of Clinical Medicine, University of Oslo 4Population Health Science, Bristol Medical School, University of Bristol, UK. 5Centre for PanorOmic Sciences, The University of Hong Kong, Hong Kong SAR, China 6State Key Laboratory for Cognitive and Brain Sciences, The University of Hong Kong, Hong Kong SAR, China 7Medical Research Council Integrative Epidemiology Unit at the University of Bristol, Bristol, UK #Joint first authors *Joint senior authors Corresponding author: David M Evans Email: [email protected] Telephone: +61 7 3443 7347 Address: University of Queensland Diamantina Institute Level 7, 37 Kent St Translational Research Institute Woolloongabba, QLD 4102 Australia bioRxiv preprint doi: https://doi.org/10.1101/2020.02.21.959114; this version posted February 25, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity.
    [Show full text]
  • Collection of Problems on Smarandache Notions", by Charles Ashbacher
    Y ---~ y (p + 1,5(p + 1)) p p + 1 = Ex-b:u.s "1:T::n.i."'I7ex-lI!d.~ Pre._ •Va.:U. ~996 Collect;io:n. of Problern..s O:n. Srn..a.ra:n.dache N"o"i;io:n.s Charles Ashbacher Decisio:n.rn..a.rk 200 2:n.d A:v-e. SE Cedar Rapids, IA. 52401 U"SA Erhu.s U":n.i"V"ersity Press Vall 1996 © Cha.rles Ashbacher & Erhu.s U":n.i"V"ersity Press The graph on the first cover belongs to: M. Popescu, P. Popescu, V. Seleacu, "About the behaviour of some new functions in the number theory" (to appear this year) . "Collection of Problems on Smarandache Notions", by Charles Ashbacher Copyright 1996 by Charles Ashbacher and Erhus University Press ISBN 1-879585-50-2 Standard Address Number 297-5092 Printed in the United States of America Preface The previous volume in this series, An Introduction to the Smarandache Function, also by Erhus Cniversity Press, dealt almost exclusively with some "basic" consequences of the Smarandache function. In this one, the universe of discourse has been expanded to include a great many other things A Smarandache notion is an element of an ill-defined set, sometimes being almost an accident oflabeling. However, that takes nothing away from the interest and excitement that can be generated by exploring the consequences of such a problem It is a well-known cliche among writers that the best novels are those where the author does not know what is going to happen until that point in the story is actually reached.
    [Show full text]
  • Arxiv:1608.06086V1 [Math.NT]
    POWER OF TWO AS SUMS OF THREE PELL NUMBERS JHON J. BRAVO, BERNADETTE FAYE AND FLORIAN LUCA Abstract. In this paper, we find all the solutions of the Diophantine equation a Pℓ +Pm +Pn =2 , in nonnegative integer variables (n,m,ℓ,a) where Pk is the k-th term of the Pell sequence Pn n≥0 given by P0 = 0, P1 = 1 and Pn+1 =2Pn+Pn−1 for all n 1. { } ≥ MSC: 11D45, 11B39; 11A25 Keywords: Diophantine equations, Pell numbers, Linear forms in logarithm, reduction method. 1. Introduction The Pell sequence P is the binary reccurent sequence given by P = 0, P =1 { n}n≥0 0 1 and Pn+1 = 2Pn + Pn−1 for all n 0. There are many papers in the literature dealing with Diophantine equations≥ obtained by asking that members of some fixed binary recurrence sequence be squares, factorials, triangular, or belonging to some other interesting sequence of positive integers. For example, in 2008, A. Peth˝o[18] found all the perfect powers (of exponent larger than 1) in the Pell sequence. His result is the following. Theorem 1 (A. Peth˝o, [18]). The only positive integer solutions (n, q, x) with q 2 of the Diophantine equation ≥ q Pn = x are (n, q, x) = (1, q, 1) and (7, 2, 13). That is, the only perfect powers of exponent larger than 1 in the Pell numbers are 2 P1 =1 and P7 = 13 . The case q = 2 had been treated earlier by Ljunggren [13]. Peth˝o’s result was rediscovered by J. H.
    [Show full text]
  • Rapid Multiplication Modulo the Sum and Difference of Highly Composite Numbers
    MATHEMATICS OF COMPUTATION Volume 72, Number 241, Pages 387{395 S 0025-5718(02)01419-9 Article electronically published on March 5, 2002 RAPID MULTIPLICATION MODULO THE SUM AND DIFFERENCE OF HIGHLY COMPOSITE NUMBERS COLIN PERCIVAL Abstract. We extend the work of Richard Crandall et al. to demonstrate how the Discrete Weighted Transform (DWT) can be applied to speed up multiplication modulo any number of the form a b where p is small. In ± p ab particular this allows rapid computation modulo numbers of thej form k 2n 1. Q · ± In addition, we prove tight bounds on the rounding errors which naturally occur in floating-point implementations of FFT and DWT multiplications. This makes it possible for FFT multiplications to be used in situations where correctness is essential, for example in computer algebra packages. 1. Introduction In their seminal paper of 1994, Richard Crandall and Barry Fagin introduced the Discrete Weighted Transform (DWT) as a means of eliminating zero-padding when performing integer multiplication modulo Mersenne numbers [2]. While this does not give any improvement in the order of the multiplication, it nevertheless cuts the transform length (and thus time and memory) in half. For these reasons the DWT has become a fixture of the search for Mersenne primes [10]. By using the same form of irrational-base representation as is used for Mersenne numbers, we can in fact eliminate the zero-padding when working modulo a b ± provided that p ab p is sufficiently small that we have enough precision. Essen- tially, as with Mersennej numbers, we choose the base so that the reduction modulo xn 1 implicitQ in the FFT multiplication turns into a reduction modulo a b.
    [Show full text]
  • (Premium Quality) 50.00 $ 1 PL30 Sentence Analysis
    CR25.PW Upper Elementary Classroom (9-12) - Premium Quality Updated DecemBer 30, 2019 Qty Code Name Price LANGUAGE 1 PL29 Sentence Analysis: First Chart and Box (Premium Quality) $ 50.00 1 PL30 Sentence Analysis Chart (Premium Quality) $ 130.00 1 PL31 Set of Arrows and Circles for Sentence Analysis (Premium Quality) $ 15.00 MATHEMATICS & GEOMETRY 1 M29-5 Large Fraction Skittles (Set Of 5) $75.00 1 M100 Metal Inscribed And Concentric Figures $110.00 1 PM01 Binomial Cube (Premium Quality) $70.00 1 PM02 Trinomial Cube (Premium Quality) $110.00 1 PM34 Cut Out Fraction Labels (1-10) $160.00 1 PM15 Cut Out Fraction Labels (12-24) $75.00 1 M135 Equivalent Figure Material $275.00 1 M145 Theorem of Pythagoras $180.00 1 PM04.A Algebraic Trinomial Cube (Premium Quality) $110.00 1 PM42 Fraction Circles (Premium Quality) $200.00 1 PM43 Fraction Squares (Premium Quality) $200.00 1 PM55 Banker Game (Premium Quality) $55.00 1 PM58 Cubing Material (Premium Quality) $850.00 1 PM59 Colored Counting Bars (Premium Quality) $200.00 1 PM60 Flat Bead Frame (Premium Quality) $50.00 1 PM61 Numbers with Symbols (Premium Quality) $110.00 1 PM62 Small Square Root Board (Premium Quality) $56.00 1 PM64 Checker Board Beads (Premium Quality) $85.00 1 PM65 Checker Board Number Tiles w/ Box (Premium Quality) $37.00 1 PM66 Five Yellow Prisms for Volume with Wooden Cubes (Premium Quality) $85.00 1 PM67 Yellow Triangles for Area (Premium Quality) $115.00 1 PM68 Long Division Material (Premium Quality) $190.00 1 PM72 Power of Two Cube (Premium Quality) $70.00 1 PM72.A Power
    [Show full text]