61. Puzzle 37.-  Set Of Even Numbers { Ai } Such That Every Ai + Aj + 1 Is Prime ( As above, some prime numbers are formed more than once by this sequence, 2 perfect solutions were found containing 17 numbers (generating 136 unique http://www.primepuzzles.net/puzzles/puzz_037.htm

Puzzles Puzzle 37.- I have found the following set of 14 even numbers with the property described above: These are the 91 primes formed: Questions: a) Find another set with 14 members b) Does the before mentioned set of 14 members accept another valid member? c) Find a larger set d) j) Solution Jack Brennen "Carlos, I have some results on your latest puzzle!!! Well, this is a very nice work! By the way Jack has pointed out that my solution - contains a condition that I was not aware of it: all the primes produced by this solution are different !!...[91/91, CBRF] Obviously, my next question - after the Brennen work - is this: I have found a solution to d) that can be a good starting point to improve: The 14 members set is this: CBRF, 6/2/99 Jack Brennen (8/2/99) has produced a larger solution to d): a set with 15 members and 105/105 distinct primes. This is the amazing set: Wilfred W h iteside has found better (larger) solutions to c) of this puzzle. At 3/05/99 he wrote:

62. Surfing The Net With Kids: Web Search Results An organized search for Mersenne prime numbers. Free software provided. largest known prime, mersenne, Mersenne, Mersenne prime, perfect number prime . http://www.surfnetkids.com/related.php?t=Prime Numbers&c=/primenumbers.htm

63. 40th Mersenne Prime Is Discovered 40th Mersenne prime Yields Largest Known perfect Number. perfect numbers andLarge numbers of Digits. Although other media may report on this find, http://members.aol.com/DrMWEcker/Mersenne.htm

64. Trailpost 2: Properties Of Prime Numbers An example of how to find perfect numbers using Mersenne primes is given below.You can find perfect numbers by using the formula (2n 1) * 2n-1, http://www.cs.usask.ca/resources/tutorials/csconcepts/1999_7/tutorial/trail/tp02

Trailpost 2: Prime Numbers What You Will Learn What prime numbers are. Mersenne Primes and perfect numbers. Sieve of Eratosthenes(a.k.a. Tex the Saskatchewan Ranger). Learn It Natural numbers are either prime or composite numbers. A prime number is a natural number that can only be divided by one and itself. In other words, it has exactly two factors. For example the number can only be divided by and , so is a prime number. has the factors and so is a composite number. Numbers that have more than two factors are composite numbers. Marin Mersenne 1588 - 1648 A special type of prime is called a Mersenne prime. Mersenne primes are calculated using the formula n . Any prime number calculated using the formula is a Mersenne prime. For example, the number is a Mersenne prime since and is a prime number. Note that the formula does not always produce prime numbers. For example, , which is not a prime number. Mersenne primes are very rare. In fact there are only known Mersenne primes as of January 1998. Mersenne primes can be used to calculate a special type of number called Perfect numbers. These are numbers whose factors when added together equal the number. For example the number

65. Numbers Carmichael numbers behave like prime numbers with respect to the most useful Less than perfect numbers are called deficient, too perfect abundant. http://www.tanyakhovanova.com/Numbers/numbers.html

What kinds of numbers are out there? Enter your number Index Definitions Other Links Index abundant amicable apocalyptic powers aspiring ... weird Definitions abundant Definiton: The number n is abundant if the sum of all its positive divisors except itself is more than n. They are abundant above perfection, not to mention deficiency. See perfect and deficient numbers. First ten: There are 2487 abundant numbers below 10,000. amicable Definiton: The number n is amicable if it belongs to an amicable pair. Two numbers n and m are called an amicable pair if the sum of all positive divisors of n is equal to the sum of all positive divisors of m and both are equal to n + m. It all started with perfect numbers which are amicable with themselves. Those numbers adopted social virtues and qualities; for the parts of each of them have the power to generate the other. See also social numbers. First ten: There are 10 amicable numbers below 10,000. apocalyptic powers Definiton: The number n is called an apocalyptic power if 2 n contains the digits 666 (in decimal).

66. Knight's Tour Art The number 2 matches all other prime numbers because it has a 1 at the beginning For additional analysis on primes, Mersenne primes, or perfect numbers, http://www.borderschess.org/KTprimes.htm

From Knight Moves to Primes From looking at the moves of the knight on the chessboard, I wanted to find out what the slope angle in degrees was for the hypotenuse of the (2, 1) right triangle made by the knight. I remembered from my old trigonometry days that I could use the Pythagorean Theorem to solve this problem. I also decided to find the angles for other types of similar triangles. I ultimately ended up with a neat summation formula. Select the formula below to see my math analysis. After finding the angles in degrees from the formula, I plotted them on the following graph: Afterwords, I added x,y coordinates to the squares where the angled lines intersected whole integer pairs. I began to realize that the coordinates were the same numbers that represent the factors for each integer. I then made a mirror image of the chart and replaced all the x,y coordinates of each square with ones and zeros. Before showing the binary and prime chart, here is a chart created by writing down the first 11 x,y coordinates of each angle starting with (0,0) from the previous chart. This new chart can be used as a multiplication chart. Increasing the length of the angled slopes in the previous chart will reveal additional integers and their factors that can be used to increase the size of the multiplicaton chart. Since the factorial chart also looked like binary, I went ahead and created a new binary chart and overlaid the same angles (only inverted and rotated 90 degrees counter-clockwise) previously discovered. The chart ultimately reveals that all primes fit within a specific pattern made by the angles of right triangles found by my summation formula.

67. Jossey-Bass::Prime Numbers: The Most Mysterious Figures In Math patents on prime numbers. Pépin’s test for Fermat numbers. perfect numbers.odd perfect numbers. perfect, multiply. permutable primes. http://www.josseybass.com/WileyCDA/WileyTitle/productCd-0471462349,descCd-tableO

By Keyword By Title By Author By ISBN By ISSN Shopping Cart My Account Help Contact Us ... Mathematics Prime Numbers: The Most Mysterious Figures in Math Related Subjects Popular Interest Mathematics Historical Mathematics Related Titles Mathematics Speed Mathematics: Secrets Skills for Quick Calculation (Paperback) by Bill Handley GAMM - Mitteilungen (Journal) ZAMM - Zeitschrift fuer Angewandte Mathematik und Mechanik (Journal) Mathematische Nachrichten (Journal) The Liar Paradox and the Towers of Hanoi: The Ten Greatest Math Puzzles of All Time (Paperback) by Marcel Danesi by Stephen J. Micklo Random Structures and Algorithms (Journal) Mathematics Prime Numbers: The Most Mysterious Figures in Math David Wells ISBN: 0-471-46234-9 Hardcover 288 pages May 2005 US $24.95 Add to Cart Other Available Formats: E-Book Description Table of Contents Read Excerpt: Table of Contents (PDF) Read Excerpt: Index (PDF) Author Information Acknowledgments. Introduction. Entries A to Z. abc conjecture.

68. Perfect Numbers At the heart of every perfect number is a Mersenne prime. be a perfect numberwith (2c+1 1) being the embedded Mersenne prime. http://www.grc.nasa.gov/WWW/K-12/Numbers/Math/Mathematical_Thinking/prfctno.htm

Proficiency Tests Mathematical Thinking in Physics Aeronauts 2000 CONTENTS Introduction Fermi's Piano Tuner Problem How Old is Old? If the Terrestrial Poles were to Melt... ... A Note on the Centrifugal and Coriolis Accelerations as Pseudo Accelerations - PDF File On Expansion of the Universe - PDF File Perfect Numbers - A Case Study Perfect numbers are those numbers that equal the sum of all their divisors including 1 and excluding the number itself. Most numbers do not fit this description. At the heart of every perfect number is a Mersenne prime. All of the other divisors are either powers of 2 or powers of 2 times the Mersenne prime. Let's examine the number 496 - one of the known perfect numbers. In order to demonstrate that 496 is a perfect number, we must show that 496 = (the sum of all its divisors including 1 and excluding 496) We might just start by dividing and working out the divisors the long way. Or, we might begin by noting that, in the notation that includes a Mersenne prime, x 31.

69. HAKMEM -- NUMBER THEORY, PRIMES, PROBABILITY -- DRAFT, NOT YET PROOFED The number of primes in 10^12 + 1 to 10^12 + 10018 is 335; the prime numbertheorem predicts 363 in The first four perfect numbers are 6, 28, 496, 8128. http://www.inwap.com/pdp10/hbaker/hakmem/number.html

Beeler, M., Gosper, R.W., and Schroeppel, R. HAKMEM . MIT AI Memo 239, Feb. 29, 1972. Retyped and converted to html ('Web browser format) by Henry Baker, April, 1995. NUMBER THEORY, PRIMES, PROBABILITY Previous Up Next ITEM 28 (Schroeppel): After about 40 minutes of run time to verify the absence of any non-trivial factors less than 235, the 125th Mersenne number, was factored on Tuesday, January 5, 1971, in 371 seconds run time as follows: John Brillhart at the University of Arizona had already done this. M137 was factored on Friday, July 9, 1971 in about 50 hours of computer time: Current prime records H.B. ITEM 29 (Schroeppel): For a random number X, the probability of its largest prime factor being greater than sqrt(X)=X^(1/2) is ln 2. less than X^(1/3) is about 4.86%. This suggests that similar probabilities are independent of X; for instance, the probability that the largest prime factor of X is less than X^(1/20) may be a fraction independent of the size of X. RELEVANT DATA: ([] denote the expected value of adjacent entries.) RANGE COUNT CUMULATIVE SUM OF COUNT 10^12 to 10^6 7198 [6944] 10018 10^6 to 10^4 2466 2820 10^4 to 10^3 354 402 [487] 10^3 to 252 40 48 ;252 = 10^2.4 252 to 100 7 8 100 to 52 1 1 ;52 = 10^1.7 51 to 1

70. Perfect Number Numbers Prime Mersenne Greater Factors perfect Number numbers prime Mersenne Greater Factors Economy. http://www.economicexpert.com/a/Perfect:number.html

var GLB_RIS='http://www.economicexpert.com';var GLB_RIR='/cincshared/external';var GLB_MMS='http://www.economicexpert.com';var GLB_MIR='/site/image';GLB_MML='/'; document.write(''); document.write(''); document.write(''); document.write(''); A1('s',':','html'); Non User A B C ... Home First Prev [ 1 Next Last In mathematics, a perfect number is an integer which is the sum of its proper positive divisor s, excluding itself. Thus, is a perfect number, because 1, 2 and 3 are its proper positive divisors and 1 + 2 + 3 = 6. The next perfect number is  = 1 + 2 + 4 + 7 + 14. The next perfect numbers are and (sequence in OEIS ). These first four perfect numbers were the only ones known to the ancient Greeks The Greek mathematician Euclid discovered that the first four perfect numbers are generated by the formula 2 n n for n for n for n for n Noticing that 2 n prime number in each instance, Euclid proved that the formula 2 n n n Ancient mathematicians made many assumptions about perfect numbers based on the four they knew. Most of the assumptions were wrong. One of these assumptions was that since 2, 3, 5, and 7 are precisely the first four primes, the fifth perfect number would be obtained when n = 11, the fifth prime. However, 2

71. CORE.NU -- Fonts Other perfect numbers do exist, but not within the 1–26 series. Furthermore,you ll find nine prime numbers in the alphabet (1–26) http://www.core.nu/perfectfont.html

The Perfect Font - (Oct 02) Pierre de Fermat, mathematician in France (1601-1665), proved that the number 26 has a unique property: it is the only number that is squeezed in between a square and a cube: 5 in square: 5 3 in cube: 3 So, there is something special about the number 26. Of course it is: 26 is also the number of letters in our alphabet. Naturally, the fontgrid for each letter should contain 26 cells. To the best of my knowledge it's not possible to build a grid with 26 cells, other than the 2x13: And that is just silly, you can't build decent font with that. The closest number to 26 is 25 and 27. So which one should I use? 25 has two divisors: 1 and 5. The sum of 1 and 5 is 6. The fact that the alphabet has 6 vowels, qualifies the 5x5 grid for The Perfect Font. 27 has three divisors: 1, 3 and 9. Add them up and you get 13. 13 is a prime number*, which makes this grid a pure choice. The 3x3x3 grid is a hexagon which means 6 outline corners; 6 vowels. [*A prime number can be divided, only by itself and by 1.] Now that these two grids are established, it's time to get busy:

72. Positive Integral Divisors A prime number is a positive number whose only divisors are 1 and itself. If you know your number is not a perfect square, then you can count in pairs. http://www.algebra-online.com/positive-integral-divisors-1.htm

Our software, Algebra Buster solves any algebra problem you enter (including all the problems found in tutorials below and much more! ). It gives you all the solution steps and clear explanations. Click here for demo or to find out more about this incredible program! Positive Integral Divisors 1 In order to understand positive integral divisors, we need to understand divisors, prime numbers and prime factorizations, and how each of these parts of number theory work with each other. Divisors, Primes, and Prime Factorization If a whole number is divided by a whole number and the quotient is a whole number, then the two numbers are called divisors of the originals number. For example, since 24 ÷ 6 = 4 and all the numbers are whole numbers, we can say that 6 is a divisor of 24. Also notice that 4 is a divisor of 24 too, since we can change the order to get 24 ÷ 4 = 6. Therefore, both 4 and 6 are divisors of 24. Other divisors of 24 include 1, 2, 3, 8, 12, and 24. A prime number is a positive number whose only divisors are 1 and itself. The prime factorization of a number is written as the product of the prime numbers that make up that number.

73. Number Theory - Numericana An odd perfect number with k prime factors can t exceed 24k Nielsen 2003.The question of finding an odd perfect number, or showing that none exist, http://home.att.net/~numericana/answer/numbers.htm

home index units counting ... physics Final Answers , Ph.D. The number 1 is not prime , as definitions are chosen to make theorems simple. Composite numbers are not prime, but the converse need not be true... Two prime numbers whose sum is equal to their product. Gaussian integers : Factoring into primes on a two-dimensional grid. The least common multiple may be obtained without factoring into primes. Modular Arithmetic may be used to find the last digit(s) of very large numbers. Powers of ten expressed as products of two factors without zero digits Divisibility by 7, 13, and 91 (or by B -B+1 in base B). Standard Factorizations : n + 4 is never prime for n because Linear equation in integers : Use Bezout's theorem and/or Euclid's algorithm Lucky 7's . Any integer divides a number composed of only 7's and 0's. The number of divisors of an integer Perfect numbers and Mersenne primes Binary and/or hexadecimal numeration for floating-point numbers as well. Fast exponentiation by repeated squaring. Partition function . How many collections of positive integers add up to 15? A Lucas sequence whose oscillations never carry it back to -1.

74. Colours Of Numbers The colour of a prime number is determined by its remainder from dividing by 12 as All known perfect numbers 28, 496, 8128, except 6 are green. http://www.hermetic.ch/misc/numcol.htm

Colours of Numbers by Karl Palmen The Rules for Colouring Numbers Fascinating Facts and Trivia Challenges I discovered a way of colouring the natural numbers that I have found very fascinating. I use following eight colours: black, red, green, yellow, blue, magenta, cyan and white . (Before printing this page on a colour printer see the note at the bottom.) It started years ago when I realised that those numbers that can be expressed as the sum of just two squares (1, 2, 4, 5, 8, 9, 10, 13 etc.) contain all their multiplication products (e.g., 2x5=10). This arises as a consequence of De Moivre's theorem in complex numbers. I became quite fascinated by these numbers and worked out a large number of them. I soon suspected a relationship between such numbers and the remainders of their prime factors divided by 4. From the geometry of the complex plane I discovered a similar set of numbers. These are the numbers expressable as the sum of two squares and their geometric mean (1, 3, 4, 7, 9, 12, 13 etc.). These too contain their multiplication products (e.g., 3x4=12). I soon suspected a relationship between such numbers and the remainders of their prime factors divided by 3. These considerations eventually inspired me to find my way of colouring numbers. The numbers that are the sum of two squares are either

75. Euclid's Elements, Book IX, Proposition 36 If 2p – 1 is a prime number, then (2p – 1) 2p1 is a perfect number. The Mersenne primes 2p – 1 corresponding to these four perfect numbers are 3, 7, http://aleph0.clarku.edu/~djoyce/java/elements/bookIX/propIX36.html

Proposition 36 If as many numbers as we please beginning from a unit are set out continuously in double proportion until the sum of all becomes prime, and if the sum multiplied into the last makes some number, then the product is perfect. Let as many numbers as we please, A, B, C, and D, beginning from a unit be set out in double proportion, until the sum of all becomes prime, let E equal the sum, and let E multiplied by D make FG. I say that FG is perfect. For, however many A, B, C, and D are in multitude, take so many E, HK, L, and M in double proportion beginning from E. Therefore, ex aequali A is to D as E is to M. Therefore the product of E and D equals the product of A and M. And the product of E and D is FG, therefore the product of A and M is also FG. VII.14 VII.19 Therefore A multiplied by M makes FG. Therefore M measures FG according to the units in A. And A is a dyad, therefore FG is double of M. But M, L, HK, and E are continuously double of each other, therefore E, HK, L, M, and FG are continuously proportional in double proportion. Subtract from the second HK and the last FG the numbers HN and FO

76. Euclid's Elements, Book VII Definition 22 A perfect number is that which is equal to the sum its Properties of prime numbers are presented in propositions VII.30 through VII.32. http://aleph0.clarku.edu/~djoyce/java/elements/bookVII/bookVII.html

Table of contents Definitions Propositions Guide Definitions Definition 1 A unit is that by virtue of which each of the things that exist is called one. Definition 2 A number is a multitude composed of units. Definition 3 A number is a part of a number, the less of the greater, when it measures the greater; Definition 4 But parts when it does not measure it. Definition 5 The greater number is a multiple of the less when it is measured by the less. Definition 6 An even number is that which is divisible into two equal parts. Definition 7 An odd number is that which is not divisible into two equal parts, or that which differs by a unit from an even number. Definition 8 An even-times even number is that which is measured by an even number according to an even number. Definition 9 An even-times odd number is that which is measured by an even number according to an odd number. Definition 10 An odd-times odd number is that which is measured by an odd number according to an odd number. Definition 11 A prime number is that which is measured by a unit alone. Definition 12 Numbers relatively prime are those which are measured by a unit alone as a common measure.

77. Mathematics What is the relation between perfect numbers and primes? A number is perfect whenit equals the sum of its divisors. 6 is the first perfect number, http://www.braungardt.com/Mathematica/Index_Mathematica.htm

Mathematics Home About Me Seminars Contact Me ... In German Philosophy Psychoanalysis Religion Theologie Theology Lacan Physics Mathematics Psychotherapy Thinking Miscellaneous Quotes about Mathematics Paul Erdös , the most prolific mathematician and problem-solver of the 20th century. He proved, for instance, that there is always a prime between n and The Prime Number Theorem describes the distribution of prime numbers. Euclid could prove that there is an infinite number of primes, but their location can only be predicted by statistical means, as an approximation. The Fundamental Theorem of Arithmetic and its proof. It states that every positive integer can be written as a product of prime numbers in a unique way. Georg Cantor, who discovered the transfinite numbers. The Beginnings of Set Theory. This text describes in (almost) plain English the history of the problems that led to Cantor's formulation of set theory. Transfinite numbers: aleph c aleph . Cantors argument: The cardinality of real numbers (c, for continuum, or aleph ) is infinitely larger than the countable infinity of natural numbers ( aleph ). You can find a good exposition of this argument on

78. Perfect Numbers new Mersenne prime was found, thus bringing the total number of known perfectnumbers It is not known if there are infinitely many perfect numbers, http://www.newton.dep.anl.gov/newton/askasci/1995/math/MATH064.HTM

Ask A Scientist Mathematics Archive Perfect numbers Author: kantars How are perfect numbers generated? Response #: 1 of 2 Author: tee If K is a prime and M(K) = 2^K-1 is also a prime (now called a Mersenne prime) then P(K) =2^(K-1)*M(K) is a perfect number (the sum of all of its proper divisors is equal to P(K)). Response #: 2 of 2 Author: rcwinther In fact, Euler proved that ALL *even* perfect numbers MUST be of the form given in response #1. Just recently a new Mersenne prime was found, thus bringing the total number of known perfect numbers (if I remember correctly) to 33. It is not known if there are infinitely many perfect numbers, nor it is known whether there are any odd perfect numbers. (However, in 1973 it was proven that, if there are, they must be larger than 10^50.) Back to Mathematics Ask A Scientist Index NEWTON Homepage Ask A Question ... NEWTON is an electronic community for Science, Math, and Computer Science K-12 Educators. Argonne National Laboratory, Division of Educational Programs, Harold Myron, Ph.D., Division Director.

79. Atlas: Prime Gaps Modulo A Perfect Number By Rahul Athale The difference between any two consecutive prime numbers is called a We consider the distribution of prime gaps modulo six, which is a perfect number http://atlas-conferences.com/c/a/k/l/24.htm

Atlas home Conferences Abstracts about Atlas Journées Arithmétiques XXIII July 6-12, 2003 University of Graz and University of Technology of Graz Graz, Styria, Austria Organizers S. Frisch, A. Geroldinger, P. Grabner, F. Halter-Koch, C. Heuberger, G. Lettl, R. Tichy View Abstracts Conference Homepage Prime Gaps Modulo a Perfect Number by Rahul Athale Research Institute for Symbolic Computation (RISC), Hagenberg, Austria The difference between any two consecutive prime numbers is called a prime gap. We consider the distribution of prime gaps modulo six, which is a perfect number with respect to the usual definition (A natural number is called a perfect number if the sum of all its divisors, excluding itself, is equal to the number.). We call six a perfect number due to the property of the resulting distribution of prime gaps modulo six: The number of prime gaps congruent to zero modulo six is approximately same as the number of prime gaps not congruent to zero modulo six. This also substantiates the claim made in a recent Science Update on the Nature web site; statistically the difference between consecutive prime gaps is rarely a multiple of six. We also give the estimate of the distribution of prime gaps modulo six using Hardy-Littlewood k-tuple conjecture.

80. Prime Formulas Fermat numbers On the way to find a prime formula, in the 1640s FERMAT perfect numbers We call a number perfect, if the sum of its divisors not http://heja.szif.hu/ANM/ANM-000926-A/anm000926a/node3.html

HEJ, HU ISSN 1418-7108 Manuscript no.: ANM-000926-A Prime formulas The great mathematicians for centuries were trying to give formulas, which would always produce primes, or at least infinitely many primes. For the second part of this question a nice answer was given by the following theorem, which analyses the occurence of prime numbers in arithmetic sequences. T HEOREM 6. (B Y D IRICHLET) Let and be integer numbers, for which gcd . In this case the sequence produces infinitely many primes R EMARKS As special cases of Theorem 6., there are infinitely many primes in the form We can rephrase the results as follows: the polynomial with gcd produces infinitely many primes. In this context we can formulate some other questions, e.g. a) Is there a polynomial in the form , which produces infinitely many primes? b) Is there a polynomial which always produces prime numbers? In the first case it is easy to prove, that necessary conditions are the irreducibility of the polynomial and gcd , but the complete answer is still unknown. To question b), for

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