DOCSLIB.ORG

The Unfathomable World of Amazing Numbers

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Unfathomable World of Amazing Numbers 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.
Load more

Recommended publications
  • Asymptotic Formulae for the Number of Repeating Prime Sequences Less Than N
    Notes on Number Theory and Discrete Mathematics Print ISSN 1310–5132, Online ISSN 2367–8275 Vol. 22, 2016, No. 4, 29–40 Asymptotic formulae for the number of repeating prime sequences less than N Christopher L. Garvie Texas Natural Science Center, University of Texas at Austin 10100 Burnet Road, Austin, Texas 78758, USA e-mail: [email protected] Received: 21 April 2016 Accepted: 30 October 2016 Abstract: It is shown that prime sequences of arbitrary length, of which the prime pairs, (p; p+2), the prime triplet conjecture, (p; p+2; p+6) are simple examples, are true and that prime sequences of arbitrary length can be found and shown to repeat indefinitely. Asymptotic formulae compa- rable to the prime number theorem are derived for arbitrary length sequences. An elementary proof is also derived for the prime number theorem and Dirichlet’s Theorem on the arithmetic progression of primes. Keywords: Primes, Sequences, Distribution. AMS Classification: 11A41. 1 Introduction We build up the sequences of primes using a variant of the sieve process devised by Eratosthenes. In the sieve process the integers are written down sequentially up to the largest number we wish to check for primality. We first remove from consideration every number of the form 2n (n≥2), find the smallest number >2 that was not removed, that is 3 (the next prime), and similarly now remove all numbers of the form 3n (n≥2). As before find again the next number not removed, that is the next prime 5. The process is continued up to a desired prime Pr.
    [Show full text]
  • Interprime 198 Yacht Marine for the Maintenance of Above Water Areas
    Interprime 198 Yacht Marine For the maintenance of above water areas. PRODUCT DESCRIPTION A quick drying, one pack primer. Interprime 198 is surface tolerant, compatible with most substrates and can be overcoated with a wide range of finishes. PRODUCT INFORMATION Colour CPA097-White, CPA098-Grey, CPA099-Red Finish Matt (ISO 2813 : 1978) Specific Gravity 1.25 Volume Solids 41% average (ISO 3233 : 1998) Typical Shelf Life 1 yrs VOC (As Supplied) 506 g/lt Unit Size 5 Lt 20 Lt DRYING/OVERCOATING INFORMATION Drying 5ºC 15ºC 23ºC 35ºC Touch Dry [ISO] 3hrs 1.5hrs 60mins 30mins Hard Dry [ISO] 24hrs 10hrs 4hrs 2hrs Overcoating Substrate Temperature 5ºC 15ºC 23ºC 35ºC Overcoated By Min Max Min Max Min Max Min Max Interlac 665 6hrs 3days 4hrs 2.5days 2hrs 2days 60mins 24hrs Interprime 198 3hrs ext 2hrs ext 60mins ext 30mins ext Interthane 990 24hrs 7days 18hrs 7days 12hrs 7days 6hrs 3days Pre-Kote 24hrs 5days 18hrs 4days 12hrs 3days 8hrs 24hrs APPLICATION AND USE Preparation Use in accordance with the standard Global Yacht specifications. All surfaces to be coated should be clean, dry and free from contamination. High pressure fresh water wash or fresh water wash as appropriate, to remove all oil, grease, soluble contaminates and other detrimental foreign matter. Maintenance & Repair: Prepare area to be repaired to a minimum of St2 (ISO 8501-1:1998). Higher levels of surface preparation by abrasive blasting to Sa 2 (ISO 8501 - 1:1998) or hydroblasting to HB2M (International Hydroblasting Standards), will enhance product performance. Feather or chip back surrounding area to a sound edge.
    [Show full text]
  • Number 73 Is the 37Th Odd Number
    73 Seventy-Three LXXIII i ii iii iiiiiiiiii iiiiiiiii iiiiiiii iiiiiii iiiiiiii iiiiiiiii iiiiiiiiii iii ii i Analogous ordinal: seventy-third. The number 73 is the 37th odd number. Note the reversal here. The number 73 is the twenty-first prime number. The number 73 is the fifty-sixth deficient number. The number 73 is in the eighth twin-prime pair 71, 73. This is the last of the twin primes less than 100. No one knows if the list of twin primes ever ends. As a sum of four or fewer squares: 73 = 32 + 82 = 12 + 62 + 62 = 12 + 22 + 22 + 82 = 22 + 22 + 42 + 72 = 42 + 42 + 42 + 52. As a sum of nine or fewer cubes: 73 = 3 13 + 2 23 + 2 33 = 13 + 23 + 43. · · · As a difference of two squares: 73 = 372 362. The number 73 appears in two Pythagorean triples: [48, 55, 73] and [73, 2664, 2665]. Both are primitive, of course. As a sum of three odd primes: 73 = 3 + 3 + 67 = 3 + 11 + 59 = 3 + 17 + 53 = 3 + 23 + 47 = 3 + 29 + 41 = 5 + 7 + 61 = 5 + 31 + 37 = 7 + 7 + 59 = 7 + 13 + 53 = 7 + 19 + 47 = 7 + 23 + 43 = 7 + 29 + 37 = 11 + 19 + 43 = 11 + 31 + 31 = 13 + 13 + 47 = 13 + 17 + 43 = 13 + 19 + 41 = 13 + 23 + 37 = 13 + 29 + 31 = 17 + 19 + 37 = 19 + 23 + 31. The number 73 is the 21st prime number. It’s reversal, 37, is the 12th prime number. Notice that if you strip off the six triangular points from the 73-circle hexagram pictured above, you are left with a hexagon of 37 circles.
    [Show full text]
  • Sds Interprime 198 Grey International Paint
    CPA098_D4 Safety Data Sheet INTERPRIME 198 GREY Sales Order: Sales Order Bulk Sales Reference No.: CPA098 SDS Revision Date: 02/28/2019 SDS Revision Number: D4-3 1. Identification of the preparation and company 1.1. Product identifier Product Identity INTERPRIME 198 GREY Bulk Sales Reference No. CPA098 1.2. Relevant identified uses of the substance or mixture and uses advised against Intended Use Paints and Coatings 1.3. Details of the supplier of the safety data sheet Company Name International Paint LLC Manufacturer: Akzo Nobel Coatings International Paint 6001 Antoine Drive Houston, Texas 77091 National Supplier: Akzo Nobel Coatings Ltd. 110 Woodbine Downs Blvd. Unit #4 Etobicoke, Ontario Canada M9W 5S6 +1 (800) 618-1010 Emergency CHEMTREC (800) 424-9300 International Paint (713) 682-1711 Customer Service International Paint LLC (800) 589-1267 Fax No. (800) 631-7481 2. Hazard identification of the product 2.1. Classification of the substance or mixture Flam. Liq. 3;H226 Flammable liquid and vapor. Skin Irrit. 2;H315 Causes skin irritation. Eye Irrit. 2;H319 Causes serious eye irritation. Skin Sens. 1;H317 May cause an allergic skin reaction. Carc. 2;H351 Suspected of causing cancer. Aquatic Chronic 3;H412 Harmful to aquatic life with long lasting effects. 2.2. Label elements Using the Toxicity Data listed in section 11 & 12 the product is labelled as follows. 1/10 CPA098_D4 Warning. H226 Flammable liquid and vapour. H315 Causes skin irritation. H317 May cause an allergic skin reaction. H319 Causes serious eye irritation. H351 Suspected of causing cancer. H412 Harmful to aquatic life with long lasting effects.
    [Show full text]
  • Results for Wieferich Primes 2
    Results for Wieferich Primes N. A. Carella Abstract: Let v 2 be a fixed integer, and let x 1 and z x be large numbers. ≥ ≥ ≥ The exact asymptotic formula for the number of Wieferich primes p, defined by vp−1 ≡ 1 mod p2, in the short interval [x,x + z] is proposed in this note. The search conducted on the last 100 years have produced two primes p<x = 1015 such that 2p−1 1 mod p2. ≡ The probabilistic and theoretical information within predicts the existence of another base v = 2 prime on the interval [1015, 1040]. Furthermore, a result for the upper bound on the number of Wieferich primes is used to demonstrate that the subset of nonWieferich primes has density 1. AMS Mathematical Subjects Classification: Primary 11A41; Secondary 11B25. Keywords: Distribution of Prime, Wieferiech prime, Finite Rings. Contents 1 Introduction 3 1.1 SummaryofHeuristics. .. .. .. .. .. .. .. .. 3 1.2 ResultsInShortIntervals . ... 4 1.3 AverageOrder .................................. 4 1.4 Guide ....................................... 5 2 Basic Analytic Results 6 2.1 SumsAndProductsOverThePrimes . 6 2.2 TotientsFunctions ............................... 7 2.3 Sums Of Totients Functions Over The Integers . ...... 7 2.4 Sums Of Totients Functions Over The Primes . ..... 9 2.5 Sums Of Totients Functionsc Over Subsets Of Integers . ......... 9 arXiv:1712.08166v2 [math.GM] 5 May 2018 2.6 Problems ..................................... 11 3 Finite Cyclic Groups 13 3.1 MultiplicativeOrders. 13 3.2 MaximalCyclicSubgroups . 14 4 Characteristic Functions 15 4.1 Characteristic Functions Modulo Prime Powers . ........ 15 4.2 Characteristic Functions Modulo n ....................... 16 4.3 Problems ..................................... 17 5 Equivalent Exponential Sums 17 1 results for wieferich primes 2 6 Upper Bound For The Main Term 20 7 Evaluations Of The Main Terms 21 7.1 SumsOverThePrimes.............................
    [Show full text]
  • Sequences of Primes Obtained by the Method of Concatenation
    SEQUENCES OF PRIMES OBTAINED BY THE METHOD OF CONCATENATION (COLLECTED PAPERS) Copyright 2016 by Marius Coman Education Publishing 1313 Chesapeake Avenue Columbus, Ohio 43212 USA Tel. (614) 485-0721 Peer-Reviewers: Dr. A. A. Salama, Faculty of Science, Port Said University, Egypt. Said Broumi, Univ. of Hassan II Mohammedia, Casablanca, Morocco. Pabitra Kumar Maji, Math Department, K. N. University, WB, India. S. A. Albolwi, King Abdulaziz Univ., Jeddah, Saudi Arabia. Mohamed Eisa, Dept. of Computer Science, Port Said Univ., Egypt. EAN: 9781599734668 ISBN: 978-1-59973-466-8 1 INTRODUCTION The definition of “concatenation” in mathematics is, according to Wikipedia, “the joining of two numbers by their numerals. That is, the concatenation of 69 and 420 is 69420”. Though the method of concatenation is widely considered as a part of so called “recreational mathematics”, in fact this method can often lead to very “serious” results, and even more than that, to really amazing results. This is the purpose of this book: to show that this method, unfairly neglected, can be a powerful tool in number theory. In particular, as revealed by the title, I used the method of concatenation in this book to obtain possible infinite sequences of primes. Part One of this book, “Primes in Smarandache concatenated sequences and Smarandache-Coman sequences”, contains 12 papers on various sequences of primes that are distinguished among the terms of the well known Smarandache concatenated sequences (as, for instance, the prime terms in Smarandache concatenated odd
    [Show full text]
  • Wieferich Primes and Period Lengths for the Expansions of Fractions
    314 MATHEMATICSMAGAZINE Wieferich Primesand PeriodLengths for the Expansionsof Fractions GENE GARZA JEFF YOUNG University of Montevallo Montevallo, Al 35115 [email protected] It is well known that some decimal expansions terminate, while others repeat, at least eventually, in patterns,which may be short or lengthy (we shall call this repeating pattern the period of the expansion). Here we will extend some known results while exploring expansions of fractions in any base. Our goal will be to find a formula for the length of the period of such expansions. The interested reader is referred to the recent award-winning article by Jones and Pearce, who show how to display such decimal expansions graphically [3]. We will consider both the expansions of (the reciprocals of) primes and of com- posites. It would seem that the easier part of this problem would be that of primes. However, there are difficulties/anomalies among primes that make it hard to find a for- mula that works in all cases. The most interesting such case is that of Wieferich primes, whose reciprocals are characterizedby expansions whose periods are the same length as the periods of their squares. For example, the length of the period of 1/1093 is 1092 which is the same as that of 1/10932. This, as we shall see, is not normally the case. For someone seeking a simple formula, this is bad news. However, as our table at the end shows, Wieferich primes are quite rare. Preliminaries Let's review what is meant by the expansion of a fraction and, in particular,the decimal expansion of a fraction.
    [Show full text]
  • On Hardy's Apology Numbers
    ON HARDY’S APOLOGY NUMBERS HENK KOPPELAAR AND PEYMAN NASEHPOUR Abstract. Twelve well known ‘Recreational’ numbers are generalized and classified in three generalized types Hardy, Dudeney, and Wells. A novel proof method to limit the search for the numbers is exemplified for each of the types. Combinatorial operators are defined to ease programming the search. 0. Introduction “Recreational Mathematics” is a broad term that covers many different areas including games, puzzles, magic, art, and more [31]. Some may have the impres- sion that topics discussed in recreational mathematics in general and recreational number theory, in particular, are only for entertainment and may not have an ap- plication in mathematics, engineering, or science. As for the mathematics, even the simplest operation in this paper, i.e. the sum of digits function, has application outside number theory in the domain of combinatorics [13, 26, 27, 28, 34] and in a seemingly unrelated mathematical knowledge domain: topology [21, 23, 15]. Pa- pers about generalizations of the sum of digits function are discussed by Stolarsky [38]. It also is a surprise to see that another topic of this paper, i.e. Armstrong numbers, has applications in “data security” [16]. In number theory, functions are usually non-continuous. This inhibits solving equations, for instance, by application of the contraction mapping principle because the latter is normally for continuous functions. Based on this argument, questions about solving number-theoretic equations ramify to the following: (1) Are there any solutions to an equation? (2) If there are any solutions to an equation, then are finitely many solutions? (3) Can all solutions be found in theory? (4) Can one in practice compute a full list of solutions? arXiv:2008.08187v1 [math.NT] 18 Aug 2020 The main purpose of this paper is to investigate these constructive (or algorith- mic) problems by the fixed points of some special functions of the form f : N N.
    [Show full text]
  • Interprotect (Professional) High Performance Epoxy Primer for Steel, Aluminium and Wood
    Interprotect (Professional) High Performance Epoxy Primer for Steel, Aluminium and Wood PRODUCT DESCRIPTION Quick drying, easy to apply epoxy primer for high performance protection. * Excellent anticorrosive protection * Antifouling tie-coat for use over epoxy primers * Good abrasion resistance PRODUCT INFORMATION Colour YPA400*-White, YPA403*-Grey Finish Matt Specific Gravity 1.4 Volume Solids 45% Mix Ratio 3:1 by volume , Converter/Curing Agent - YPA404* Typical Shelf Life 2 yrs VOC (As Supplied) 464 g/lt Unit Size 750 ml 2.5 Lt 5 Lt DRYING/OVERCOATING INFORMATION Drying 5ºC 15ºC 23ºC 35ºC Touch Dry [ISO] 4hrs 2hrs 1hrs 30mins Immersion 24hrs 18hrs 12hrs 8hrs Pot Life 10hrs 5hrs 3hrs 2hrs Overcoating Substrate Temperature 5ºC 15ºC 23ºC 35ºC Overcoated By Min Max Min Max Min Max Min Max Interfill 830 36hrs 3mths 24hrs 3mths 24hrs 3mths Interfill 830 Fast Cure 36hrs 3mths 24hrs 3mths 24hrs 3mths International/Interlux Antifouling 24hrs 48hrs 10hrs 20hrs 6hrs 16hrs 4hrs 10hrs Interprotect (Professional) 10hrs 6mths 5hrs 6mths 3hrs 6mths 2hrs 6mths Interspeed Extra Strong 24hrs 48hrs 10hrs 20hrs 6hrs 16hrs 4hrs 10hrs Interspeed Ultra 24hrs 48hrs 10hrs 20hrs 6hrs 16hrs 4hrs 10hrs Micron 55 24hrs 48hrs 10hrs 20hrs 6hrs 16hrs 4hrs 10hrs Micron 66 24hrs 48hrs 10hrs 20hrs 6hrs 16hrs 4hrs 10hrs Micron CSC 24hrs 48hrs 10hrs 20hrs 6hrs 16hrs 4hrs 10hrs Micron Extra 24hrs 48hrs 10hrs 20hrs 6hrs 16hrs 4hrs 10hrs Micron Extra (Professional) 24hrs 48hrs 10hrs 20hrs 6hrs 16hrs 4hrs 10hrs Perfection Undercoat 24hrs 7days 16hrs 4days 12hrs 2days 6hrs 1days Polyurethane Basecoat 24hrs 7days 16hrs 4days 12hrs 2days 6hrs 1days Trilux 24hrs 48hrs 10hrs 20hrs 6hrs 16hrs 4hrs 10hrs Trilux 33 24hrs 48hrs 10hrs 20hrs 6hrs 16hrs 4hrs 10hrs Uni-Pro 24hrs 48hrs 10hrs 20hrs 6hrs 16hrs 4hrs 10hrs Note:If maximum overcoating time is exceeded, sand with 180-220 grade wet or dry paper.
    [Show full text]
  • Elementary Proof That an Infinite Number of Cullen Primes Exist
    Elementary Proof that an Infinite Number of Cullen Primes Exist Stephen Marshall 21 February 2017 Abstract This paper presents a complete proof of the Cullen Primes are infinite, even though only 16 of them have been found as of 21 Feb 2017. We use a proof found in Reference 1, that if p > 1 and d > 0 are integers, that p and p+ d are both primes if and only if for integer m: m = (p-1)!( + ) + + We use this proof for d = to prove the infinitude of Cullen prime numbers. The author would like to give many thanks to the authors of 1001 Problems in Classical Number Theory, Jean-Marie De Koninck and Armel Mercier, 2004, Exercise Number 161 (see Reference 1). The proof provided in Exercise 6 is the key to making this paper on the Cullen Prime Conjecture possible. Introduction James Cullen, (19 April 1867 – 7 December 1933) was born at Drogheda, County Louth, Ireland. At first he was educated privately, then he studied pure and applied mathematics at the Trinity College, Dublin for a while, he went to 1 Mungret College, Limerick, before deciding to become a Jesuit, studying in England in Mansera House, and St. Mary's, and was ordained as a priest on 31 July 1901. In 1905, he taught mathematics at Mount St. Mary's College in Derbyshire and published his finding of what is now known as Cullen numbers in number theory. In number theory of prime numbers, a Cullen number is a natural number of the form n Cn = n2 + 1 Cullen numbers were first studied by James Cullen in 1905.
    [Show full text]
  • X22a CFHT Megaprime U-Band Source Catalog of the AKARI North Ecliptic Pole Field
    2020 Autumn Annual Meeting, Astronomical Society of Japan X22a CFHT Megaprime u-band Source Catalog of the AKARI North Ecliptic Pole Field Ting-Chi Huang (SOKENDAI, ISAS/JAXA), Hideo Matsuhara (SOKENDAI, ISAS/JAXA), To- motsugu Goto (NTHU), Tetsuya Hashimoto (NTHU), Seong-Jin Kim (NTHU), Nagisa Oi (TUS), Hyunjin Shim (KNU), Ho Seong Hwang (KASI), Yoshiki Toba (Kyoto Univ.), Daryl Joe D. Santos (NTHU), and Toshinobu Takagi (JSF) AKARI infrared (IR) space telescope carried out a survey in the North Ecliptic Pole (NEP) field using its Infrared Camera (IRC), and detected more than 100,000 IR sources. AKARI IRC’s 9 filters continuously cover the wavelength from near- to mid-IR, making AKARI unique in comparison with other IR space telescopes like Spitzer or WISE. However, the research in the AKARI NEP field was limited due to the lack of sufficient optical and ultraviolet (UV) observations. Recently, we have performed an observation in the AKARI NEP field using Subaru HSC, and obtained deep optical images. Now we further provide the near-UV (u band) catalog from Canada-France-Hawaii Telescope (CFHT) Megaprime to assist the precious AKARI data. The observation was conducted in 7 nights under the Queued Service Observations mode in 2015 and 2016, and covered a 3.6-deg2 area. The data were processed by the Elixir pipeline, and the mosaic image was created by the AstrOmatic software. Compared with previous work, this work not only extends the area coverage to the whole AKARI NEP field, but also reaches a deeper imaging, which is a significant improvement for us to estimate photometric redshifts of galaxies or study stellar populations..
    [Show full text]
  • Mathematical Constants and Sequences
    Mathematical Constants and Sequences a selection compiled by Stanislav Sýkora, Extra Byte, Castano Primo, Italy. Stan's Library, ISSN 2421-1230, Vol.II. First release March 31, 2008. Permalink via DOI: 10.3247/SL2Math08.001 This page is dedicated to my late math teacher Jaroslav Bayer who, back in 1955-8, kindled my passion for Mathematics. Math BOOKS | SI Units | SI Dimensions PHYSICS Constants (on a separate page) Mathematics LINKS | Stan's Library | Stan's HUB This is a constant-at-a-glance list. You can also download a PDF version for off-line use. But keep coming back, the list is growing! When a value is followed by #t, it should be a proven transcendental number (but I only did my best to find out, which need not suffice). Bold dots after a value are a link to the ••• OEIS ••• database. This website does not use any cookies, nor does it collect any information about its visitors (not even anonymous statistics). However, we decline any legal liability for typos, editing errors, and for the content of linked-to external web pages. Basic math constants Binary sequences Constants of number-theory functions More constants useful in Sciences Derived from the basic ones Combinatorial numbers, including Riemann zeta ζ(s) Planck's radiation law ... from 0 and 1 Binomial coefficients Dirichlet eta η(s) Functions sinc(z) and hsinc(z) ... from i Lah numbers Dedekind eta η(τ) Functions sinc(n,x) ... from 1 and i Stirling numbers Constants related to functions in C Ideal gas statistics ... from π Enumerations on sets Exponential exp Peak functions (spectral) ..
    [Show full text]