Third Edition Number Treasury Investigations, Facts and Conjectures about More than 100 Number Families Margaret J Kenney • Stanley J Bezuszka Boston College, Massachusetts, USA World Scientific Published by World Scientific Publishing Co Pte Ltd Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE Library of Congress Cataloging-in-Publication Data Kenney, Margaret J Number treasury : investigations, facts, and conjectures about more than 100 number families / by Margaret J Kenney (Boston College, USA) and Stanley J Bezuszka (Boston College, USA) 3rd edition pages cm Includes bibliographical references and index ISBN 978-9814603683 (hardcover : alk paper) ISBN 978-9814603690 (softcover : alk paper) Numeration Mathematical recreations I Bezuszka, Stanley J., 1914–2008 II Title III Title: Number treasury three QA141.K46 2015 513.5 dc23 2014040942 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Copyright © 2015 by World Scientific Publishing Co Pte Ltd Printed in Singapore Contents Foreword xi A Perfect Number of Investigations 28 = + + + + 14 Numbers Based on Divisors and Proper Divisors Positive Integers Divisors, Multiples and Proper Divisors Prime and Composite Numbers Sieve of Eratosthenes Prime Factorization Property (Fundamental Theorem of Arithmetic) Testing for Primes Divisors of an Integer, GCD, and LCM Relatively Prime and Euler φ Numbers Abundant, Deficient, and Perfect Numbers Sums and Differences of Abundant and Deficient Numbers Products of Abundant and Deficient Numbers Multiples of Perfect Numbers Consecutive Integers and Abundant Numbers Abundant Numbers as Sums of Abundant Numbers Powers of Primes and Deficient Numbers Even and Odd Integers, Even Perfect Numbers, Mersenne Primes Multiply Perfect Numbers Almost Perfect Numbers Semiperfect Numbers Weird Abundant Numbers Operations on Semiperfect Numbers Primitive Semiperfect Numbers Amicable Numbers Imperfectly Amicable Numbers Sociable Numbers and Crowds 39 39 39 40 41 42 44 45 46 48 49 53 54 55 56 56 58 61 62 62 63 64 64 65 66 67 June 4, 2015 8:57 Number Treasury3 9.75in x 6.5in b2012-fm page viii Practical Numbers 68 Baselike Numbers 71 Plane Figurate Numbers 73 Polygons Figurate Numbers Triangular Numbers Operations on Triangular Numbers Perfect Numbers, Triangular Numbers and Sums of Cubes Pascal’s Triangle Triangle Inequality Numbers Rectangular Numbers Square Numbers Sums of Square Numbers Positive Square Pair Numbers Bigrade Numbers Pythagorean Triples Primitive Pythagorean Triples Congruent Numbers Fermat’s Last Theorem Happy Numbers Operations on Happy Numbers Happy Number Words Repeating Cycles Patterns in Squares of 1, 11, 111, Squarefree Numbers Tetragonal Numbers Pentagonal Numbers Hexagonal Numbers Recursion and Figurate Numbers Remainder Patterns in Figurate Numbers Gnomic Numbers Lo-Shu Magic Square; Male and Female Numbers Solid Figurate Numbers Polyhedra and Solid Figurate Numbers Pyramidal Numbers Tetrahedral Numbers, Triangular Pyramidal Numbers Square Pyramidal Numbers Pentagonal Pyramidal Numbers Hexagonal Pyramidal Numbers Heptagonal and Octagonal Pyramidal Numbers 73 74 74 77 78 79 80 83 85 88 89 90 90 92 93 94 94 95 96 97 98 99 100 101 103 107 109 109 111 113 113 115 115 118 120 120 121 June 4, 2015 8:57 Number Treasury3 9.75in x 6.5in b2012-fm Star Numbers and Star Pyramidal Numbers Rectangular Pyramidal Numbers Cubic Numbers Integers as Sums and Differences of Cubic Numbers, 1729 Pythagorean Parallelepiped Numbers More Prime Connections Goldbach’s Conjectures Integers as Sums of Odd Integers Integers as Sums of Two Composite Numbers Positive Prime Pair Numbers Prime Line and Prime Circle Numbers Beprisque Numbers A Primes-Between Property Germain Primes Twin Primes Semiprimes and Boolean Integers Snowball Primes Lucky Numbers Prime and Lucky Numbers Polya’s Conjecture about Odd- and Even-Type Integers Balanced Numbers Fermat Numbers Cullen Numbers Ruth–Aaron Numbers page ix 121 122 125 126 128 131 Digital Patterns and Noteworthy Numbers Monodigit and Repunit Numbers and Langford Sequences Social and Lonely Numbers Additive Multidigital Numbers Multiplicative Multidigital Numbers Kaprekar’s Number 6174, 99 and 1089 Doubling Numbers Good Numbers Nearly Good Semiperfect Numbers Powerful Numbers Armstrong Numbers and Digital Invariant Numbers Narcissistic Numbers Additive Digital Root Numbers Additive Persistence of Integers Multiplicative Digital Root Numbers Multiplicative Persistence of Integers 131 133 134 134 136 138 138 139 140 140 142 142 144 145 147 147 148 149 151 151 153 155 156 157 159 161 163 164 165 167 167 169 170 171 June 4, 2015 8:57 Number Treasury3 9.75in x 6.5in b2012-fm page x Modest and Extremely Modest Numbers 173 Visible Factor Numbers 174 Nude Numbers 175 More Patterns and Other Interesting Numbers More Sums of Consecutive Integers Product Patterns for Consecutive Integers Consecutive Integer Divisors Consecutive Number Sums and Square Numbers Factorial Numbers, Applications and Extensions Factorial Sum Numbers and Subfactorial Numbers Hailstone and Ulam Numbers; The Collatz and Ulam Conjecture Palindromic Numbers Creating Palindromic Numbers Palindromic Number Words and Curiosities Palindromic Numbers and Figurate Numbers Palindromic Primes and Emirps Honest Numbers Bell Numbers Catalan Numbers Fibonacci Numbers Lucas Numbers Tribonacci Numbers Tetranacci Numbers Phibonacci Numbers Survivor Numbers or U-Numbers or Ulam Numbers Tautonymic Numbers Lagado Numbers 177 177 181 183 184 185 188 189 191 193 194 196 196 197 198 200 203 206 207 208 209 209 210 211 Recommended Readings 215 Glossary of Numbers 217 Solutions to Investigations 225 Solutions to Exercises 249 Index 305 June 4, 2015 8:57 Number Treasury3 9.75in x 6.5in b2012-fm Foreword The gift of numbers like the gift of fire has made the world much brighter Stanley J Bezuszka (1914–2008) Introduction Time and numbers were born together Time is a measure of change, and numbers express that measure There is an awesome mystery that surrounds the all-pervasive dimension of time Likewise, there is an awesome mystery and fascination about numbers that attract distinguished mathematics researchers as well as imaginative amateurs the world over Teachers who pursue numbers and the rich history of numbers with their classes can provide students with an understanding that mathematics is a collaborative effort that has been nurtured by individuals and groups representing many cultures and periods of history Students can learn to become accomplished investigators, to make discoveries, and contribute to a branch of mathematics that is vibrant and motivating Number Treasury3 has evolved in order to serve as a catalyst for those who ascribe to this point of view Details Number Treasury3 is a broadening and update of Number Treasury2 The book contains information about more than 100 families of positive integers Brief historical notes often accompany the descriptions and examples of the number families Exercises for each major family are provided to stimulate insight Some exercises contain problems that are thought provokers to be resolved simply with paper and pencil; others should be tackled with calculator in hand so that lengthier computations can be managed with ease and take the results to a higher level of understanding Still other problems are intended for more extensive exploration with the use of computer software In some instances it is helpful to model problems with hands-on materials page xi June 4, 2015 8:57 Number Treasury3 9.75in x 6.5in b2012-fm The emphasis in Number Treasury3 is on doing rather than proving However, the reader is urged to think critically about situations, to provide reasoned explanations, to make generalizations and to formulate conjectures The book begins with a chapter of Investigations These are principally stand-alone activities that represent content drawn from the Chapters through of the book Their purpose is to set the tone of the book and to stimulate student reflection and research in a variety of areas In fact, throughout the book, the reader will find numerous open-ended problems This book also contains detailed solutions to the Exercises and Investigations A Glossary and Index are provided for quick access to information References and recommended readings are supplied so that teachers and students can use this book as a stepping stone to more concentrated study Who Uses Number Treasury3 This book is written for teachers and students For teachers Number Treasury3 is a resource for instructional preparation and problems, together with snapshots of mathematical history intended for teachable moments For students who are engaged in learning about number families and who are assigned problems, projects and papers, Number Treasury3 is a useful source of ideas and topics The mix of discussion with examples and illustrations is intended to serve as a writing model for the student Both audiences should think critically about the content, provide carefully reasoned explanations, make generalizations, and form conjectures Who Is Involved The first edition was completed with the able assistance of six Boston College graduates and undergraduates: Jeanne Cavanaugh, James Cavanaugh, Claudia Katze, Stephen Kokoska, Jill Nille, and Jonathan Smith Seven Boston College graduates and undergraduates were indispensable in the production of the second edition Special thanks and grateful appreciation go to Joan Martin for her thoughtful content and style suggestions, editorial advice and word processing skills; to Cynthia Tahlmore, Geraldine Mele, and Erin Mitchell for computer graphics and word processing assistance; to Allyson Russo, Shannon Toomey, and Megan Mazzara for problem solutions The third edition has been completed by the surviving original author with the invaluable assistance and perseverance of Geraldine Mele who offered not only content suggestions but who also especially contributed word processing, computer graphics and style expertise Sincere gratitude and appreciation is also extended to Joan Martin for her careful review of the manuscript page xii June 4, 2015 8:48 Number Treasury3 9.75in x 6.5in b2012-ch01 Chapter A Perfect Number of Investigations 28 = + + + + 14 A GREAT discovery solves a great problem but there is a grain of discovery in any problem George Polya (1887–1985) What are They? The Investigations that follow are a set of stand-alone activities Each Investigation focuses on at least one number family or topic relating to numbers All but six of the Investigations are described on one page Share with students that the problems in an Investigation are intended to be challenging in many ways: • The time needed to complete an Investigation may vary and exceed the time required to finish a typical homework assignment • The computation necessary to bring closure may be lengthy and demanding — even with the use of technology • The amount of writing, discussing, explaining and illustrating may be more than anticipated Teacher Tips The Investigations are listed in ascending order of difficulty in the table on the next two pages There are three levels of difficulty represented in the 28 Investigations that can be assigned individually or adapted for group work The lowest level consists of the first eight Investigations that have the least prerequisites The middle level consists of the next nine Investigations and requires more use of abstract reasoning and familiarity with algebraic expressions The final 11 Investigations challenge the student to persist and probe more deeply in order to complete the work There is also a column in the table naming the most significant prerequisite(s) needed by the student to understand and carry out the work in each Investigation The teacher may choose to provide additional content background for some specific Investigations prior to assigning them Assign the Investigations as extended homework or as in-class work Some Investigations call for the preparation of page June 4, 2015 8:48 Number Treasury3 9.75in x 6.5in b2012-ch01 Number Treasury3 reports Thus, students may need further directions, especially about the kind of resources available for them to use Students should know the Internet is an excellent resource, and that it should be used appropriately as they compile their reports Finally, pages noted in the Prerequisites column refer to related material contained in Chapters through Page Investigation 5,6 Footsteps of Lagrange Trying Trapezoids Hexagons in Black & White 8,9 Marble Art 10 Honest Number Hunt 11 Seeking Honesty in Numbers 12 Geoboard Journeys 13 Mysterious Mountains & Binary Trees Fermat Factorings Factor Lattices A Juggling Act The Super Sum Conjecturing with Pascal Pythagorean Triple Pursuits Pentagonal Play Triangular Number Turnarounds Centered Triangular Numbers 14 15,16 17 18 19,20 21 22 23 24 25,26 27 28,29 30 31 Catalan Capers Highly Composite Numbers Tower of Hanoi & the Reve’s Puzzle Perfect Number Patterns Crisscross Cubes Prerequisites & Text reference Square numbers, p 85 Triangular numbers & trapezoids, p 74 Familiarity with recursive & explicit formulas, p 107 Figurate numbers, following directions, pp 74–111 Counting in a language, searching resources, p 197 Groups work together to organize their data, p 197 Trial/Error pursuit, link Catalan & Pascal, p 200 Doing & arranging sketches, p 200 Prime factorization, p 42 LCM, prime factorization, p 46 Describing systematically, p 46 Reasoning with patterns, p 74 Articulating patterns, p 79 Evaluating expressions, p 90 Squares & triangles within, p 101 Visualizing triangles, p 80 Making algebraic generalizations, p 74 Connect algebra & geometry, p 201 Counting divisors, p 45 Recursive actions & thinking, p 56 Using logs to count, p 59 Perimeter, area, volume, make generalizations, p 125 page June 4, 2015 Number Treasury3 8:57 9.75in x 6.5in b2012-exerises Solutions to Exercises page 297 297 EXERCISE 125 a) 55, reversal b) 121, reversals c) 121, reversals d) 121, reversals e) 1111, reversals f) 4884, reversals g) 1111, reversals h) 4884, reversals i) 44044, reversals j) 1111, reversals k) 4884, reversals l) 44044, reversals m) 8813200023188, 24 reversals n) 88555588, 11 reversals o) 233332, reversals p) 233332, reversals q) 233332, reversals r) 99099, reversals s) 99099, reversals t) 67276, reversals Several answers are possible a) 3.3, 4.4, 5.5, 6.6 b) 45.54, 13.31, 22.22, 15.51 c) 256.652, 897.798, 543.345, 666.666 a) 3.3, reversal b) 13.31, reversals c) 15.51, reversals d) 1887.7881, reversals e) 39.93, reversals f) 11.11, reversals g) 1476.6741, reversals h) 35.53, reversals i) 329.923, reversals j) 59.95, reversal Of the 90 two-digit numbers 10–99, 53 require reversal, 22 require reversals, require reversals, require reversals, require reversals, and 2, namely 89 and 98, require 24 reversals to reach a palindrome EXERCISE 126 Lulu, Lee, Faith, Molly, Aika, Guri are palindromic names Results will vary December None 2222, 2332, 2442, 2552, 2662 111 11211 1121211 112121211 11212121211 11111111 1122222211 112233332211 11223344332211 11111 1122211 112232211 11223232211 1122323232211 (a) (b) (c) EXERCISE 127 66, 171, 595, 666, 3003, 5995, 8778 A fairly large triangular number that is also palindromic is 61728399382716 222 = 484, 1012 = 10201, 1112 = 12321 1013 = 1030301, 1113 = 1367631, 10013 = 1003003001 June 4, 2015 Number Treasury3 8:57 9.75in x 6.5in b2012-exerises page 298 Number Treasury3 298 a) 698896, yes d) 637832238736, yes b) 1048576, no e) 4461012876321, no c) 104060401, yes f) 4099923883299904, yes EXERCISE 128 191, 313, 353, 373, 383 a)13 and 31, 17 and 71, 37 and 73 79 and 97 b) 113 and 311, 107 and 701, 149 and 941, 157 and 751, 167 and 761, 179 and 971 c) 1033 and 3301, 1009 and 9001, 1021 and 1201, 1031 and 1301, 1061 and 1601, 1069 and 9601 d) 10061 and 16001, 10067 and 76001, 10069 and 96001, 10079 and 97001, 10091 and 19001, 10151 and 15101 199, 919 and 991, 337, 733 and 373 1009 1021 1031 1033 1061 1069 1091 1097 1109 1151 1153 1181 1193 1201 1213 1217 1229 1231 1237 1249 1259 1279 1283 1301 1381 1399 1409 1429 1439 1453 1471 1487 1511 1523 1559 1583 1597 1601 1619 1657 1723 1733 1741 1753 1789 1811 1831 1847 1879 1901 1913 1933 1949 1979 1103 1223 1321 1499 1669 1867 EXERCISE 129 B5 = 52 B6 = 203 B7 = 877 B8 = 4,140 B9 = 21,147 B10 = 115,9575 B11 = 678,570 B12 = 4,213,597 Given: lines of poetry Same letter in a column means these lines rhyme Line Line Line Line a a a a a a a b a a b a a b a a b a a a a a b c a a b b a b a c a b a b a b c a a b b a a b b c a b c b a b c c a b c d Let the customers be A B C D Let the cars be marked car 1, car 2, car 3, car Then we have 15 arrangements displayed Car Car Car Car ABCD — — — ABC D — — ABD C — — ACD B — — BCD A — — AB C D — AB CD — — AC B D — AC BD — — June 4, 2015 8:57 Number Treasury3 9.75in x 6.5in b2012-exerises Solutions to Exercises page 299 299 Car AD AD A A A A Car B BC BC BD B B Car C — D C CD C Car — — — — — D Let the donuts be J, C, H and the plates be P1 , P2 , and P3 There are arrangements P1 JCH JC JH CH J P2 H C J C P3 H Let the donuts be J, C, H , O and the plates be P1 , P2 , P3 , and P4 There are 15 possibilities as enumerated in and EXERCISE 130 The total number of ways C4 = 14 1 ×5 ×2 ×1 ×1 + + + = 14 = C4 1 14 × 14 × × × × 14 + + + +14 = 42 = C5 C6 = 132 C7 = 429 C8 = 1,430 C9 = 4,862 C11 = 58,786 Six people: C10 = 16,796 June 4, 2015 8:57 Number Treasury3 9.75in x 6.5in b2012-exerises Number Treasury3 300 Eight people: Let the dancers from shortest to tallest be 1, 2, 3, 4, 5, There are arrangements 456 356 346 246 256 123 124 125 135 134 Let the dancers from shortest to tallest be 1, 2, 3, 4, 5, 6, 7, There are 14 arrangements 5678 1234 4678 1235 4578 1236 4568 1237 3678 1245 3578 1246 3568 1247 3478 1256 3468 1257 2678 1345 2578 1346 2568 1347 2478 1356 2468 1357 There is a formula for Catalan numbers: Cn = (2n)! n!(n + 1)! EXERCISE 131 F12 = 144, F13 = 233, F14 = 377, F15 = 610, F11 = 89, F16 = 987, F17 = 1597, F18 = 2584, F19 = 4181, F20 = 6765 The number of Fibonacci numbers is unlimited This follows from the definition and rule of formation above We can always add the preceding two numbers to get a third Fibonacci number page 300 June 4, 2015 Number Treasury3 8:57 9.75in x 6.5in b2012-exerises Solutions to Exercises page 301 301 Sequence: 5, 6, 11, 17, 28, 45, 73, 118, 191, 309 7th term = 73 Sum = 803 = 11 × 73 Let the first number be a, the second b Term Fibonacci-like Sequence 10 a b a+b a + 2b 2a + 3b 3a + 5b 5a + 8b 8a + 13b 13a + 21b 21a + 34b Sum = 55a + 88b = 11 × (5a + 8b) 5 10 12 20 33 54 88 143 1 1 1 F7 = 13 F8 = 21 F9 = 34 F10 = 55 F11 = 89 F12 = 144 The sum of the first n Fibonacci numbers is Fn+2 − The Fibonacci Quarterly, the official journal of the Fibonacci Association, began publishing in February 1963 with Volume 1, Number The quarterly is a rich source of information on Fibonacci numbers, Lucas numbers and related topics EXERCISE 132 26 44 73 120 196 3 3 L7 = 29 L8 = 47 L9 = 76 L10 = 123 L11 = 199 The sum of the first n Lucas numbers is Ln+2 − June 4, 2015 8:57 Number Treasury3 9.75in x 6.5in b2012-exerises Number Treasury3 302 EXERCISE 133 T11 = 274, T15 = 3,136, T19 = 35,890, a) i) 16 b) j) 10 T12 = 504, T16 = 5,768, T20 = 66,012 c) k) 11 d) l) 17 T13 = 927, T17 = 10,609, e) m) 12 f) n) 13 T14 = 1,705, T18 = 19,513, g) 14 o) 15 h) All differences are on row All differences are on row This is a rich investigation The application of the tribonacci numbers discussed here can be found in S Bezuszka, and L D’Angelo “An Application of Tribonacci Numbers.” Fibonacci Quarterly, Vol 15, No (April 1977), pp 140–144 See also Fibonacci Quarterly, Vol 1, No (October 1963), pp 71–74, where the term tribonacci (perhaps also the term tetranacci) seems to have been coined by Mark Feinberg when he was fourteen years old EXERCISE 134 Q12 = 773, Q13 = 1, 490, Q14 = 2, 872, Q11 = 401, Q15 = 5, 536, Q16 = 10, 671, Q17 = 20, 569 Q18 = 39, 648 Q19 = 76, 424, Q20 = 147, 312 b, d, e, f The first ten terms in the phibonacci number sequence are: 1, 2, 3, 5, 7, 11, 17, 23, 37, 41 For more information on phibonacci numbers, see A Bager, “Problem E2833.” American Mathematical Monthly, Vol 87: (May 1980), p 404 EXERCISE 135 a and c are survivor numbers Answers will vary Some possibilities: a) 13 + 69 = 82 b) 82 − 69 = 13 c) 11 × 57 = 627 EXERCISE 136 a) 3838 e) 240240 b) 4646 f) 999999 c) 9999 g) 251251 d) 134134 h) 365365 abc × × 11 × 13 = abc × 1001 = abcabc d) 627/11 = 57 page 302 June 4, 2015 Number Treasury3 8:57 9.75in x 6.5in b2012-exerises Solutions to Exercises a) b) c) d) e) ÷7 = ÷11 = ÷13 = 33891 65208 98527 126984 135278 3081 5928 8957 11544 12298 237 456 689 888 946 Yes, because abcabc = abc (7 × 11 × 13) so page 303 303 abcabc = abc × 11 × 13 EXERCISE 137 a) 10(13) b) 7(22) c) 4(55) = 10(22) d) 4(58) e) 4(94) f) 4(100) = 10(40) = 16(25) g) 4(115) = 10(46) h) 4(121) = 22(22) The set of Lagado numbers is generated by the formula (3n − 2) June 4, 2015 8:58 Number Treasury3 9.75in x 6.5in b2012-index Index Integers are the fountainhead of all mathematics Hermann Minkowski (1859–1930) Families of numbers are listed individually; “number” is suppressed from the listings Aaron, H., 149 abundant, 48 and amicable, 66 chains of, 50 and deficient, 51 differences of, 51 products of, 53 sums of, 49 consecutive, 55 differences of, 51 and multiples of perfect numbers, 54 and odd, 49 and practical, 70 products of, 53 and product of consecutive, 55 and semiperfect, 62 as sums of abundant, 56 weird, 63 additive digital root, 167 as a test for triangular numbers, 168 additive multidigital, 155 and factorials, 156 additive persistence, 169 algorithm for Bell numbers, 199 Catalan numbers, 201 doubling numbers, 159 Fermat factoring, 14 good numbers, 161 lucky numbers, 142 palindromes (reverse-add), 193 prime numbers, 41 reverse-subtract, reverse-add, 158 aliquot parts, 40 almost perfect, 62 type +1, 62 type −1, 62 amicable, 65 and abundant, 66 chains of, 67 and deficient, 66 digital invariant, 166 imperfectly, 66 list of, 66 arithmetic progression, 109 arithmetic sequence, 109 Arithmetica, 94 Armstrong, 165 Armstrong, M.F., 165 Babylonians, 91 Bager, A., 209 balanced, 147 baselike, 71 and practical, 71 Beiler, A.H., 151 Bell, 199 algorithm, 199 triangle, 199 Bell, E.T., 198 beprisque, 138 Bezuszka, S.J., xi bigrade, 90 binary tree, 13 binomial coefficient, 187 Boole, G., 141 boolean integer, 141 Canterbury Puzzles, 29 Canterbury Tales, 29 page 305 June 4, 2015 306 8:58 Number Treasury3 9.75in x 6.5in b2012-index Number Treasury3 Card, L., 142 Catalan, 25, 201 von Segner’s algorithm, 201 Catalan, E.C., 25, 202 centered square, 24 triangular, 24 chain of proper divisor sums, 67 with link, 67 with links, 67 with links, 67 with 28 links, 67 Chaos game, 20 Chaucer, G., 29 Chebyshev, P.L., 138 Claus, N., 28 closed form, 107 Collatz, L., 189 Collatz and Ulam Conjecture, 189 common logarithm, 30 composite, 40 endings, 42 as product of primes, 42 sums of, 134 Comti, A., 131 concave polygon, 200 congruent, 93 conjecture, 19 consecutive, 55 and abundant, 55 composites, 188 divisors, 183 and odd and even square numbers, 184 product of four and squares, 182 products and abundant numbers, 55 products and multiples of six, 55 products and semiperfect numbers, 64 and rectangular numbers, 84 squares and triangular, 88 sums for any integer least number, 180 sums of, 177 sums of 2, 3, 4, and endings, 181 three numbers and multiples of 3, 178 three numbers and square of middle, 181 and triangular, 180 convex polygon, 200 fixed, 200 co-prime, 47 relatively prime, 47 Crest of the Peacock, 34 crowds, 68 cube (hexahedron), 114 cubic, 126 differences, 126 and palindromic, 196 sums, 127 Cullen, 148 and primes, 148 Cullen J., 148 dancers standing problem, 203 decahedron, 114 decimal, palindromic, 193 defective, 48 deficient, 48 and abundant, 51 differences of, 52 and powers of primes, 56 and primes, 57 products of, 54 sums of, 50 Descartes, R., 59, 213 diagonal in Pascal’s triangle, 80 in polygons, 200 of rectangular solids, 128 digits, 34 digital amicable invariant, 166 perfect invariant, 165 recurring invariant, 165 root (additive), 167 root (multiplicative), 170 Diophantus, 94 Disquisitiones Arithmeticae, 148 divisor(s), 39 consecutive integer, 183 consecutive of a 3-digit number, 184 exact, 39 greatest common, 46 integral, 39 number of, 27 and prime factorization, 45 proper, 30, 40 semiperfect, 62 smallest number divisible by 1,2, ,10, 184 sums of, 61 dodecahedron, 114 page 306 June 4, 2015 8:58 Number Treasury3 9.75in x 6.5in b2012-index Index Donovan, S., 159 doubling, 159 algorithm for, 159 doubly perfect, 61 Doucette, J., 193 Dudeney, H., 29 edge, 113, 128 Egyptians, 91 Elements, 59 emirp, 197 equilateral triangle, 80 Eratosthenes, 41 sieve, 41 Erdăos, P., 138, 217 property on primes, 138 Euclid, 30, 59, 83 Euler ø (phi), 47, 209 Euler, L., 25, 47, 200 and Catalan, 200 even, 58 endings, 58 perfect, 59 and rectangular, 84 sums and products of, 58 sums and products of with odds, 58 as sums of luckies, 144 as sums of odds, 58 even-type, 146 excessive, 48 explicit formula, closed form, 107 exponential notation, 45 extremely modest, 173 faces, 113 factor(s), 39, 42 lattices, 15 visible, 174 factorial, 35, 185 and additive multidigital, 156 sum, 188 and triangular, 187 female, 111 Fermat, 147 primes, 147 Fermat, P de, 14, 94 and Germain, 140 factoring algorithm, 14 Last Theorem, 94, 140 page 307 307 Fibonacci, 37, 204 -like sequence, 205 sequence, 204 tree, 204 Fibonacci, Leonardo of Pisa, 37, 203 figurate, 74 and palindromic, 196 and remainder pattern, 109 fixed convex polygon, 200 formula explicit, recursive, friendly, 65 Fundamental Theorem of Arithmetic (Prime Factorization Property), 43 Gardner, M., 61 Gauss, C.F., 18, 39, 76, 147 generator, 153 geoboard, 12 geometric solid, 113 Germain prime, 139 and Fermat’s Last Theorem, 140 Germain, S., 139 GIMPS, 59 gnomic, 110 gnomon, 110 Goldbach, C., 131 conjectures, 131 Golden Ratio, 37 good, 162 algorithm, 161 nearly good semiperfect, 163 greatest common divisor, 46 Gulliver’s Travels, 212 Hailstone, 189 handshake problem, 202 happy, 94 cycles for not happy, 97 differences of, 95 products of, 96 sums of, 95 words, 96 Hardy, G.H., 27, 127 Havermann, H., 173 heptagonal pyramidal, 121 heptahedron, 114 hexagonal, 103 and nonregular, 104 June 4, 2015 8:58 Number Treasury3 308 9.75in x 6.5in Number Treasury3 and pentagonal, 105 product of odd number and integer, 105 pyramidal, 120 and square and triangular, 104 and sum of triangles, 106 and sum of squares, 106 hexahedron (cube), 114 highly composite, 27 Hindin, H., 169 additive persistence, 169 honest, 10, 197 icosahedron, 114 imperfectly amicable, 60 integers, 39 composite, 40 consecutive, 55 divisors, 183 sums of, starting with 1, 74 starting greater than 1, 177 divisor of, 39 divisors of (from prime factorization), 45 even, 58 perfect, 59 even-type, 146 multiple, 39 odd, 58 odd-type, 145 pairs of consecutive, 55 positive, 39 as sums and differences of cubic, 127 as sums of beprisque, 138 as sums of composite, 134 as sums of luckies, 144 as sums of odds, 133 as sums of semiprimes, 141 as sums as squares, 88 as sums of triangular, 77 as sums of two primes, 131 invariant, amicable digital, 166 perfect digital, 165 recurring digital, 165 isosceles triangle, 80 Jordanus de Nemore, 24 Joseph, G.G., 34 b2012-index Kaprekar, D.R., 155 6174 pattern, 157 Katagiri, Y., 175 Kronecker, L., 151 Lagado, 211 primes, 212 Lagrange, J.L., Four Squares Theorem, Langford, C.D., 152 Langford sequence, 152 least common multiple 15, 46 LeBlanc, M.-S Germain, 139 Lee, E.J., 59 Leonardo of Pisa, 37, 203 Liber Abaci, 234 Lindon, J.A., 174 linear, 83 lonely, 153 Lucas, 206 sequence, 206 Lucas, F.E.A., 28, 206 lucky, 144 algorithm, 142 and prime, 144 Madachy, J.S., 167 magic square, 33 Lo-shu (Luoshu), 111 male, 111 Mathematical Magic Show, 61 Mersenne prime, 59 Mersenne, M., 59 Minkowski, H., 305 modest, 173 extremely, 173 monodigit, 151 multidigital additive, 155 additive and multiplicative, 157 multiplicative, 156 multiperfect, 61 multiple, 39 least common, 15, 46 multiplicative digital root, 170 multidigital, 156 persistence, 171 multiply perfect, 61 Murasaki, L.S., 36 page 308 June 4, 2015 8:58 Number Treasury3 9.75in x 6.5in b2012-index Index narcissistic, 167 and Armstrong, 167 and powerful, 167 Narcissus, 167 nearly good semiperfect, 163 Nelson, C., 149 99 pattern, 158 reverse-subtract reverse-add algorithm, 158 nonahedron, 114 nonconvex polygon, 200 nude, 175 and visible factor, 174 oblong, 83 octagonal, 108 pyramidal, 121 star (stellate), 121 pyramidal, 122 octahedron, 114 odd, 58 endings, 58 sum of distinct odds, 133 sum of luckies, 145 sums and products of, 58 sums and products of with even, 58 odd-type, 145 OEIS, 171 1089 pattern, 158 reverse-subtract reverse-add algorithm, 158 order-1 chain, 67 order-2 chain, 67 order-3 chain, 68 Paganini, B.N.I., 66 palindromic, 191 and cubes, 196 curiosities, 195 decimal, 193 and figurate, 196 primes, 197 reverse-add algorithm, 193 and squares, 196 and triangular, 196 words, 191 parallelepiped Pythagorean, 128 rectangular, 128 page 309 309 parallelogram, 128 partition, 161 Pascal, B., 79 triangle, 19, 79 and factorials, 186 Peacock, G., 66 Penney, D., 149 pentagonal, 22, 101 and hexagonal, 105 nonregular array, 22 pyramidal, 120 sum of squares, 102 sum of square and triangular, sum of triangular, 102 pentahedron, 114 perfect, 30, 48 abundant, 54 almost, 62 type +1, 62 type −1, 62 chains of, 67 digital invariant, 165 doubly, 61 Euclid’s formula, 59 even, 59 list of, 60 multiples of, 54 multiply, 61 multiperfect, 61 pluperfect, 61 number of digits in, 30 odd, 59 and practical, 70 and triangular, 78 triperfect, 61 persistence additive, 169 multiplicative, 171 phi, (Euler ø), 47, 209 phibonacci, 209 plane, 83 Plato, 83, 114 plane numbers, 83 square numbers, 83 Platonic solids, 114 pluperfect, 61 Polya, G., 1, 145 conjecture, 146 polygon, 73 concave, 200 June 4, 2015 310 8:58 Number Treasury3 9.75in x 6.5in b2012-index Number Treasury3 convex, 200 diagonals of, 200 fixed convex, 200 nonconvex, 200 regular, 74 polygonal (figurate), 74 polyhedron, 8, 113 regular, 114 types, 114 Pomerance, C., 149 positive prime pair, 135 positive square pair, 89 Poulet, P., 67 powerful, 164 and narcissistic, 167 powers, and Euler, 47 of integers, definition, 45 and powerful, 164 of ten, 34 of two and sum of their proper divisors, 56 practical, 68 and abundant, 70 and baselike, 71 and perfect, 70 and powers of primes, 70 prime, 40 algorithm to find, 41 and Armstrong, 165 and boolean integers, 141 circle, 136 and deficient, 57 divisor, 42 and emirps, 197 endings of, 42 Eratosthenes’ sieve, 41 and Euler, 47 factor, 42 and Fermat, 147 Germain, 139 line, 136 triangle, 137 and lucky numbers, 144 Mersenne, 59 palindromic, 196 positive prime pair, 135 powers of and deficient numbers, 56 powers of and sum of proper divisors, 57 and practical, 70 proofs by Erdăos, 138 relatively (co-prime), 47 snowball, 142 testing for, 44 twin, 140 Prime Factorization Property (Fundamental Theorem of Arithmetic), 43 Prime95, 59 primitive semiperfect, 64 Ptolemy III, 41 pyramidal, 115 heptagonal, 121 hexagonal, 120 octagonal, 121 pentagonal, 120 rectangular, 124 square, 8, 118 star (stellate), 122 tetrahedral, 8, 115 triangular, 115 Pythagoras, 65, 91, 113 Pythagorean parallelepiped, 129 quadruple, 129 triple, 21, 90, 122 and congruent, 93 nonprimitive, 92 primitive, 21, 92 Pythagoreans, 59, 74 quadrilateral, 128 rabbit problem, 203 Ramanujan, S., 27, 127 Randle, J., 164 Recreations in the Theory of Numbers, 151 rectangle, 128 rectangular, 83, 123 and consecutive, 84 general form, 84 parallelepiped, 128 and products, 84 pyramidal, 124 similar plane, 85 solid, 128 and sums of consecutive evens, 84 and triangular, 84 recurring digital invariant, 166 3rd and 4th order, 166 recursive formula, form, 107 page 310 June 4, 2015 8:58 Number Treasury3 9.75in x 6.5in b2012-index Index redundant, 48 repunit, 151 reverse-subtract, reverse-add algorithm, 158 Reve’s Puzzle, 28 rhyming patterns problem, 198 right circular cone, 113 cylinder, 113 Robinson, J.B., 73 root additive digital, 167 multiplicative digital, 170 Ruth–Aaron pair, 150 Ruth, G.H.(B.), 149 scalene triangle, 80 semiperfect, 62 differences of, 64 and multiples of perfect, 64 nearly good semi-perfect, 163 primitive, 64 products of, 64 and products of consecutives, 64 proper divisors of, 63 sums of, 64 ways of representing semi-perfects, 63 and weird abundant, 63 semiprime, 140 and boolean integers, 141 sequence arithmetic, 109 Fibonacci, 204 Fibonacci-like, 205 Langford, 152 Lucas, 206 tetranacci, 208 tetranacci-like, 209 tribonacci, 207 tribonacci-like, 207 series, 75 side, 73 Sierpinski triangle, 20 6174 (Kaprekar’s number), 157 Sloane, N.J.A., 171 and multiplicative persistence, 171 snowball primes, 142 sociable, 67 and chains, 67 social, 153 generator of, 153 page 311 311 solid figurate, 113, 115 solids, 114 space, 115 spatial, 115 sphere, 113 center, 113 square, 85 centered, 24 and consecutive odd integers, 85 and consecutive triangular, 87 and hexagonal, 104 and palindromic, 196 patterns in 12 , 112 , 1112 , , 98 and pentagonal, 102 product of consecutive, 182 pyramidal, 8, 118 pyramidal and tetrahedral, 119 square pair, 89 sums of, 88 and tetragonal, 101 squarefree, 99 Srinivasan, A.K., 68 stamp problems, 71 star (stellate), 121 pyramidal, 121 subfactorial, 189 survivor, 209 Swift, J., 212 sympathetic, 65 Tale of Genji, 36 tautonymic, 210 Taussky-Todd, O., 177 Taylor, R., 94 Taylor, T., 66 te Riele, H.J.J., 66 term, 75 tetragon, 100 tetragonal, 100 and square, 101 and triangular, 101 tetrahedral, 115 triangular pyramidal, 115 and square pyramidal, 118 tetrahedron, 114 tetranacci, 208 -like sequence, 209 sequence, 208 Tower of Hanoi, 28 Trait´e du triangle arithm´etique, 79 June 4, 2015 312 8:58 Number Treasury3 9.75in x 6.5in b2012-index Number Treasury3 trapezoidal, triangle inequality, 81 triangular, 23, 74 and additive digital root test, 168 centered, 24 differences of, 77 factorials, 186 and hexagonal, 106 methods for getting the numbers, 74 and pentagonal, 102 and perfect, 78 products of, 77 pyramidal, 116 and rectangular, 84 and square, 87 squares of, 79 sums of, 77 and sums of consecutives starting with 1, 74 and sums of consecutives not starting with 1, 180 and sums of cubes, 79 and tetragonal, 101 and triangle inequality numbers, 82 tribonacci, 207 -like sequences, 207 sequence, 207 triperfect, 61 twin primes, 140 Ulam, 190 and hailstone, 189 Ulam, S.M., 142, 189, 210 and lucky, 144 and survivor, 209 undecahedron, 114 unit fraction, 162 vertex, 15, 73, 113 interior, 13 visible factor, 174 and nude, 175 von Segner, J.A., 201 algorithm, 201 weird abundant, 63 Wiles, A., 94 Woltman, G., 59 wondersquare, 207 word tautonyms, 210 Yu, Emperor, 111 zero, 39 page 312 ... Number treasury : investigations, facts, and conjectures about more than 100 number families / by Margaret J Kenney (Boston College, USA) and Stanley J Bezuszka (Boston College, USA) 3rd edition. .. model stand until completely dry Top and bottom view June 4, 2015 8:48 Number Treasury3 9.75in x 6.5in b2012-ch01 Number Treasury3 10 Honest Number Hunt An honest number in a language is a number. .. Consecutive Number Sums and Square Numbers Factorial Numbers, Applications and Extensions Factorial Sum Numbers and Subfactorial Numbers Hailstone and Ulam Numbers; The