THE UNFATHOMABLE WORLD OF AMAZING NUMBERS THE UNFATHOMABLE WORLD OF AMAZING NUMBERS Wallace Jacob Notion Press Old No. 38, New No. 6, McNichols Road, Chetpet, Chennai - 600 031 First Published by Notion Press 2015 Second Edition 2015 Copyright © Wallace Jacob 2015 All Rights Reserved. ISBN: 978-93-84391-32-4 This book has been published in good faith that the work of the author is original. All efforts have been taken to make the material error-free. However, the author and the publisher disclaim the responsibility. No part of this book may be used, reproduced in any manner whatsoever without written permission from the author, except in the case of brief quotations embodied in critical articles and reviews. Dedication I thank God for this book without whose wish nothing is ever possible. (Dedicated to my parents and my wife) FOREWORD It was in the year of Our Lord Nineteen Ninety that I was browsing through the previous years ISC (Indian School Certificate) Computer Science question papers and I came across questions related to Fibonacci series, Armstrong Numbers, etc. and from then on numbers have simply fascinated me. Actually we find numbers at several places, including nature. At a personal level, I feel that it may not be an exaggeration to say that life is a game of numbers. Our life is measured in age which is a number. Our success is measured by our position in the organizational hierarchy and/or society, which is again a number. Elections are a game of numbers. My experience tells me that normally people want to be at position number one. And, here we have a book devoted to numbers. Constructive criticism of the book is solicited. I have been thinking of writing a book for the past twenty-four years and although I have carried out rigorous testing and then compiled the information in the form of this book, yet there are chances that some error(s) might have crept in due to oversight or calculation mistakes or limited Foreword understanding. Therefore, I make no claim to authenticity of the material being presented in this book. The book is divided into three parts. Part A dwells on few interesting properties of few amazing numbers. Part B contains some interesting questions and Part C dwells on concepts such as Magic square, e, Pythagorean triplets, The Towers of Hanoi, etc. VIII ACKNOWLEDGEMENT I want to place on record my sincere gratitude towards: Rt. Rev. Peter Parapullil, Bishop of Jhansi; Shri. Vishwanath Mishra, Senior System Analyst, Ebix India Facility; Dr. R. K. Saxena, Institute of Management Studies, Bundelkhand University; Dr. Pankaj Atri, Institute of Management Studies, Bundelkhand University; Dr. (Mrs.) P. Puri, Institute of Management Studies, Bundelkhand University; Dr. N. P. Tolani sir, Managing Director, Tolani Shipping Company Limited; Dr. (Mrs.) Sujata Naik-Tolani, Chairperson, Tolani Shipping Company Limited; Capt. R.K.Razdan, Provost, Tolani Maritime Institute; Dr. B. K. Saxena, Principal, Tolani Maritime Institute; Prof. Indra Basu, Tolani Maritime Institute. CONTENTS Foreword VII Acknowledgements IX PART A A Tryst With Few Amazing Numbers 1 I. The Fractions 1/27 And 1/37 1 II. The Number 39 3 III. The Fraction 1/1089 4 A. Other Properties of 1089 5 IV. The Fraction 1/998999 6 A. Tribonacci Series 6 B. Lucas Series 7 C. Tetranacci Number Series 7 V. The Fraction 1/23 7 VI. The Fraction 1/47 7 VII. Hidden Fibonacci Series 8 VIII. The Fraction 2/3 8 Contents IX. The Number 36085288503684007860 9 X. The Number 1274953680 9 XI. The Numbers 31, 331, 3331, 3331 9 XII. The Number 2519 9 XIII. The Numbers 1, 8, 17, 18, 26 And 27 10 XIV. The Number 1741725 10 XV. The Number 153 12 XVI. Emirp 12 XVII. Primorial Numbers 12 XVIII. Father Prime 13 XIX. Sophie Germain Prime 13 XX. Prime Quadruplet 13 XXI. Prime Quintuplet 14 XXII. Prime Sextuplet 14 XXIII. Wilson Prime 14 XXIV. Wieferich Prime 14 XXV. Woodall Prime 15 XXVI. Euclid-Mullin Sequence 15 XXVII. The Number 314159265358979323 84626433832795028841 15 XXVIII. Circular Primes 16 XXIX. The Numbers 1487, 4817, 8147 16 XXX. The Number 1023456987896543201 16 XXXI. The Number 751 16 XXXII. Interprime Numbers and Twin Primes 17 XII Contents XXXIII. Fermat primes 19 XXXIV. Mersenne primes 19 XXXV. Prime triplets 20 XXXVI. Left-truncatable primes, Right-truncatable primes, Two-sided primes 20 XXXVII. Permutable prime 21 XXXVIII. Titanic prime, Gigantic prime, Megaprime 21 XXXIX. Cullen primes 22 XL. Another property of the number 2519 22 XLI. The number 11826 23 XLII. The numbers 567 and 854 and similar numbers 25 XLIII. The number 8549176320 70 XLIV. Double-Base Palindrome 70 XLV. Googol and Googolplex 70 XLVI. The Number 6 71 XLVII. The Number 7 71 XLVIII. The Number 8 72 XLIX. Filzian Number 72 L. The Numbers 1547 and 2196 73 Ll. Factorion 73 LII. The Number 381654729 74 LIII. Pythagorean Triplets 74 LIV. The Numbers 276, 594, 598 and 732 75 LV. The Number 51249876 75 LVI. The Number 301 76 XIII Contents LVII. The Number 45 76 LVIII. Powerful Numbers 77 LIX. Sphenic Numbers 77 LX. Pronic (Heteromecic) Numbers 77 LXI. Repunit 78 LXII. Leyland Number 78 LXIII. Lucas-Carmichael Numbers 79 LXIV. The Number 16777216 80 LXV. Triad Numbers 80 LXVI. Each Digit Only Once in Multiplication 81 LXVII. The Number 22520 81 LXVIII. The Number 73939133 81 LXIX. Pascal’s Triangle 82 LXX. Automorphic Numbers and TrimorphicNumbers 84 LXXI. Perfect Numbers 85 LXXII. Amicable Numbers 85 LXXIII. Product of Two Numbers and the Product of their Reverse 86 LXXIV. The Numbers 2178 And 21978 87 LXXV. Numbers with more than 100 Positive Divisors 88 LXXVI. Number of Letters in Numbers in the range 1 To 99 99 XIV Contents PART B Few Puzzles 103 I. Missing Symbols 103 II. Equation 104 III. Missing Pages 105 IV. Unique Magic Square 106 V. Number of Weighings 107 VI. Number of Scientists of Different Genders Puzzle 108 VII. Unique Six-Digit Number 108 VIII. The Number 6210001000 109 IX. Judging A Pattern 109 X. Angle Between Hour-Hand And Minute-Hand 110 XI. Button Restriction 111 XII. Clock Loses Time 111 XIII. Number of the Painting 112 XIV. Drainage Pipes 112 XV. Interesting Properties of a Few Numbers 113 XVI. Hidden Patterns 115 XVII. Three-Jug Puzzle 122 XVIII. Sum of all Unique Four-Digit Integers 123 XIX. An Interesting Nine-Digit Number 124 XX. Number of Squares 126 XV Contents XXI. Number of Triangles 127 XXII. Greater of Two Numbers (Power) 129 XXIII. Greater of Two Numbers (Power) 130 XXIV. Greater of Two Numbers (Multiplication) 130 XXV. Finding the Unit’s Digit (Adding the First 99 Natural Numbers) 131 XXVI. Finding The Unit’s Digit (Adding the Factorial of Integers From 1 - 10) 131 XXVII. Greater of Two Numbers 132 XXVIII. Greater of Two Numbers 132 XXIX. Equality of a Fraction and an Integer 132 XXX. Finding the Unit’s Digit 132 XXXI. Pirate Puzzle 133 XXXII. Unique 10-Digit Number 135 XXXIII. How Does the Prisoner Escape 137 XXXIV. Discovering Relationships 137 XXXV. Cryptarithms 138 XXXVI. A Fallacy 151 XXXVII. Guessing the Number left with a Player 151 XXXVIII. Measuring 9 minutes using a 4–minute hourglass and a 7–minute hourglass 152 XXXIX. Number of pages in a book 153 XL. More cryptarithms 153 XVI Contents XLI. More cryptarithms 155 XLII. More cryptarithms 156 XLIII. More cryptarithms 156 XLIV. More cryptarithms 159 XLV. More cryptarithms 169 XLVI. Mathematical symbol puzzle 175 XLVII. Birthday 175 XLVIII. Why is log10(0) = -infinity 175 XLIX. Maximum value of X x Y 176 L. x1 + x2 + x3 + x4 = 17 176 LI. Which number comes next 264 LII. Which number comes next 264 LIII. Coloured cube problem 264 LIV. Sum of angles of a triangle 268 PART C The Concept of Magic Squares 272 Alphamagic Squares 281 Templar Magic Squares 283 Concentric or Bordered Magic Squares 283 Antimagic Squares 284 A Special Array: Array 57 285 Triangular Numbers 286 The Tower of Hanoi Problem 288 The Enigma of Pythagorean Triplets 292 Using Two odd Integers Which Differ By 2 293 XVII Contents Using Two Even Integers Which Differ By 2 293 Euclid’s Method of Generating Pythagorean Triplets 294 Plato’s Method of Generating PythagoreanTriplets 295 An Interesting Pythagorean Triplet 295 What is ‘e’? 296 Applications of ‘e’ In Everyday Life 297 The Concept of Bits, Bytes, Kilobytes 301 The Concept of Leap Year (or Intercalary Year) 303 September 1752 305 What is BMI 306 Year 2038 Problem 308 What is Ramanujan’s Magic Square 310 3 x 3 magic square containing even numbers 313 Pattern in the gap between two consecutive prime numbers 316 Distribution of prime numbers 352 Epilogue 353 XVIII PART A A TRYST WITH FEW AMAZING NUMBERS There is more to numbers than meets the eye. A study of numbers can unravel many interesting facts. This book tends to explore the enigma of a few interesting numbers. I. The fractions 1/27 and 1/37 can be considered friends, because: 1/27 = .037037037... 1/37 = .027027027... The integers 27 and 37 have a unique property. For instance, if we want to determine whether 1483967893 is divisible by 27? Then starting from the unit’s place make groups of three digits. The groups will be 893, 967, 483 and 1. Now add the four groups 1 + 483 + 967 + 893. The sum obtained is 2344. On dividing this sum (2344) by 27 the remainder obtained is 22. This will also be the remainder when 1483967893 is divided by 27. The Unfathomable World of Amazing Numbers Similarly, if we want to determine whether 1483967893 is divisible by 37? Then starting from the unit’s place make groups of three digits.