|
Full Reptend Prime
In number theory, a full reptend prime, full repetend prime, proper primeDickson, Leonard E., 1952, ''History of the Theory of Numbers, Volume 1'', Chelsea Public. Co. or long prime in base ''b'' is an odd prime number ''p'' such that the Fermat quotient : q_p(b) = \frac (where ''p'' does not divide ''b'') gives a cyclic number. Therefore, the base ''b'' expansion of 1/p repeats the digits of the corresponding cyclic number infinitely, as does that of a/p with rotation of the digits for any ''a'' between 1 and ''p'' − 1. The cyclic number corresponding to prime ''p'' will possess ''p'' − 1 digits if and only if ''p'' is a full reptend prime. That is, the multiplicative order = ''p'' − 1, which is equivalent to ''b'' being a primitive root modulo ''p''. The term "long prime" was used by John Conway and Richard Guy in their ''Book of Numbers''. Confusingly, Sloane's OEIS refers to these primes as "cyclic numbers". Base 10 Base 10 ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics."German original: "Die Mathematik ist die Königin der Wissenschaften, und die Arithmetik ist die Königin der Mathematik." Number theorists study prime numbers as well as the properties of mathematical objects made out of integers (for example, rational numbers) or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory are often best understood through the study of analytical objects (for example, the Riemann zeta function) that encode properties of the integers, primes or other number-theoretic object ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modular Arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book '' Disquisitiones Arithmeticae'', published in 1801. A familiar use of modular arithmetic is in the 12-hour clock, in which the day is divided into two 12-hour periods. If the time is 7:00 now, then 8 hours later it will be 3:00. Simple addition would result in , but clocks "wrap around" every 12 hours. Because the hour number starts over at zero when it reaches 12, this is arithmetic ''modulo'' 12. In terms of the definition below, 15 is ''congruent'' to 3 modulo 12, so "15:00" on a 24-hour clock is displayed "3:00" on a 12-hour clock. Congruence Given an integer , called a modulus, two integers and are said to be congruent modulo , if is a divisor of their difference (that is, if there is an integer such that ). Congruence modu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
149 (number)
149 (one hundred ndforty-nine) is the natural number between 148 and 150. In mathematics 149 is a prime number, the first prime whose difference from the previous prime is exactly 10, an emirp, and an irregular prime. After 1 and 127, it is the third smallest de Polignac number, an odd number that cannot be represented as a prime plus a power of two. More strongly, after 1, it is the second smallest number that is not a sum of two prime powers. It is a tribonacci number, being the sum of the three preceding terms, 24, 44, 81. There are exactly 149 integer points in a closed circular disk of radius 7, and exactly 149 ways of placing six queens (the maximum possible) on a 5 × 5 chess board so that each queen attacks exactly one other. The barycentric subdivision of a tetrahedron In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
131 (number)
131 (one hundred ndthirty-one) is the natural number following 130 and preceding 132. In mathematics 131 is a Sophie Germain prime, an irregular prime, the second 3-digit palindromic prime, and also a permutable prime with 113 and 311. It can be expressed as the sum of three consecutive primes, 131 = 41 + 43 + 47. 131 is an Eisenstein prime with no imaginary part and real part of the form 3n - 1. Because the next odd number, 133, is a semiprime, 131 is a Chen prime. 131 is an Ulam number. 131 is a full reptend prime in base 10 (and also in base 2). The decimal expansion of 1/131 repeats the digits 007633587786259541984732824427480916030534351145038167938931 297709923664122137404580152671755725190839694656488549618320 6106870229 indefinitely. In the military * Convair C-131 Samaritan was an American military transport produced from 1954 to 1956 * Strike Fighter Squadron (VFA-131) is a United States Navy F/A-18C Hornet fighter squadron stationed at Naval Air Station Oceana ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
113 (number)
113 (one hundred ndthirteen) is the natural number following 112 and preceding 114. Mathematics * 113 is the 30th prime number (following 109 and preceding 127), so it can only be divided by one and itself. 113 is a Sophie Germain prime, an emirp, an isolated prime, a Chen prime and a Proth prime as it is a prime number of the form 7 × 24 + 1. 113 is also an Eisenstein prime with no imaginary part and real part of the form 3n - 1. In base 10, this prime is a primeval number, and a permutable prime with 131 and 311. *113 is a highly cototient number and a centered square number. *113 is the denominator of 355/113, an accurate approximation to . See also * 113 (other) * A113 A113 (sometimes A-113, A-1-13, A1-13 or A11-3) is an inside joke and Easter egg in media developed by alumni of California Institute of the Arts, referring to the classroom used by graphic design and character animation students. History Student ... is A Pixar recurring inside jok ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
109 (number)
109 (one hundred ndnine) is the natural number following 108 and preceding 110. In mathematics 109 is the 29th prime number. As 29 is itself prime, 109 is a super-prime. The previous prime is 107, making them both twin primes. 109 is a centered triangular number. The decimal expansion of 1/109 can be computed using the alternating series, with F(n) the n^ Fibonacci number: ::\frac=\sum_^\infty\times (-1)^=0.00917431\dots The decimal expansion of 1/109 has 108 digits, making 109 a full reptend prime in decimal. The last six digits of the 108-digit cycle are 853211, the first six Fibonacci numbers in descending order. There are exactly 109 different families of subsets of a three-element set whose union includes all three elements, 109 different loops (invertible but not necessarily associative binary operations with an identity) on six elements, and 109 squares on an infinite chessboard that can be reached by a knight within three moves. See also *109 (other) 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
97 (number)
97 (ninety-seven) is the natural number following 96 and preceding 98. It is a prime number and the only prime in the nineties. In mathematics 97 is: * the 25th prime number (the largest two-digit prime number in base 10), following 89 and preceding 101. * a Proth prime and a Pierpont prime as it is 3 × 25 + 1. * the eleventh member of the Mian–Chowla sequence. * a self number in base 10, since there is no integer that added to its own digits, adds up to 97. * the smallest odd prime that is not a cluster prime. * the highest two-digit number where the sum of its digits is a square. * the number of primes <= 29. * The numbers 97, 907, 9007, 90007 and 900007 are all primes, and they are all happy primes. However, 9000007 (read as ''nine million seven'') is |
61 (number)
61 (sixty-one) is the natural number following 60 (number), 60 and preceding 62 (number), 62. In mathematics 61 is: *the 18th prime number. *a twin prime with 59 (number), 59. *a cuban prime of the form ''p'' = , where ''x'' = ''y'' + 1. *the smallest ''proper prime'', a prime ''p'' which ends in the digit 1 in base 10 and whose reciprocal in base 10 has a repeating decimal, repeating sequence with length ''p'' − 1. In such primes, each digit 0, 1, ..., 9 appears in the repeating sequence the same number of times as does each other digit (namely, times). *the exponent of the 9th Mersenne prime. (261 − 1 = ) *the sum of two squares, 52 + 62. *a centered square number. *a centered hexagonal number. *a centered decagonal number. *the sixth alternating permutation#Related integer sequences, Euler zigzag number (or Up/down number). *a unique prime in base 14, since no other prime has a 6-digit period in base 14. *a Pillai prime since 8! + 1 is divisible by 61 but 61 is not one mor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
59 (number)
59 (fifty-nine) is the natural number following 58 and preceding 60. In mathematics Fifty-nine is the 17th prime number. The next is sixty-one, with which it comprises a twin prime. 59 is an irregular prime, a safe prime and the 14th supersingular prime. It is an Eisenstein prime with no imaginary part and real part of the form . Since is divisible by 59 but 59 is not one more than a multiple of 15, 59 is a Pillai prime. It is also a highly cototient number. There are 59 stellations of the regular icosahedron, inclusive of the icosahedron. 59 is one of the factors that divides the smallest composite Euclid number. In this case 59 divides the Euclid number 13 # + 1 = 2 × 3 × 5 × 7 × 11 × 13 + 1 = 59 × 509 = 30031. 59 is the highest integer a single symbol may represent in the Sexagesimal system. As 17 is prime, 59 is a super-prime. The number 59 takes 3 iterations of the "reverse and add" process to form the palindrome 1111. All smaller integers (1 through 58) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
47 (number)
47 (forty-seven) is the natural number following 46 and preceding 48. It is a prime number. In mathematics Forty-seven is the fifteenth prime number, a safe prime, the thirteenth supersingular prime, the fourth isolated prime, and the sixth Lucas prime. Forty-seven is a highly cototient number. It is an Eisenstein prime with no imaginary part and real part of the form . It is a Lucas number. It is also a Keith number because its digits appear as successive terms earlier in the series of Lucas numbers: 2, 1, 3, 4, 7, 11, 18, 29, 47, ... It is the number of trees on 9 unlabeled nodes. Forty-seven is a strictly non-palindromic number. Its representation in binary being 101111, 47 is a prime Thabit number, and as such is related to the pair of amicable numbers . In science * 47 is the atomic number of silver. Astronomy * The 47-year cycle of Mars: after 47 years – 22 synodic periods of 780 days each – Mars returns to the same position among the stars and is in the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
29 (number)
29 (twenty-nine) is the natural number following 28 and preceding 30. Mathematics * 29 is the tenth prime number, and the fourth primorial prime. * 29 forms a twin prime pair with thirty-one, which is also a primorial prime. Twenty-nine is also the sixth Sophie Germain prime. * 29 is the sum of three consecutive squares, 22 + 32 + 42. * 29 is a Lucas prime, a Pell prime, and a tetranacci number. * 29 is an Eisenstein prime with no imaginary part and real part of the form 3n − 1. 29 is also the 10th supersingular prime. * None of the first 29 natural numbers have more than two different prime factors. This is the longest such consecutive sequence. * 29 is a Markov number, appearing in the solutions to ''x'' + ''y'' + ''z'' = 3''xyz'': , , , , etc. * 29 is a Perrin number, preceded in the sequence by 12, 17, 22. * 29 is the smallest positive whole number that cannot be made from the numbers , using each exactly once and using only addition, subtraction, multiplication, and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
23 (number)
23 (twenty-three) is the natural number following 22 and preceding 24. In mathematics Twenty-three is the ninth prime number, the smallest odd prime that is not a twin prime. It is, however, a cousin prime with 19, and a sexy prime with 17 as well as 29. Twenty-three is also the fifth factorial prime, and the second Woodall prime. It is an Eisenstein prime with no imaginary part and real part of the form 3''n'' − 1. 23 is the fifth Sophie Germain prime and the fourth safe prime, 23 is the next to last member of the first Cunningham chain of the first kind to have five terms (2, 5, 11, 23, 47). Since 14! + 1 is a multiple of 23 but 23 is not one more than a multiple of 14, 23 is a Pillai prime. 23 is the smallest odd prime to be a highly cototient number, as the solution to ''x'' − φ(''x'') for the integers 95, 119, 143, 529. It is also a happy number in base-10. *In decimal, 23 is the second Smarandache–Wellin prime, as i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
