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Fortunate Number
In number theory, a Fortunate number, named after Reo Fortune, is the smallest integer ''m'' > 1 such that, for a given positive integer ''n'', ''p''''n''# + ''m'' is a prime number, where the primorial ''p''''n''# is the product of the first ''n'' prime numbers. For example, to find the seventh Fortunate number, one would first calculate the product of the first seven primes (2, 3, 5, 7, 11, 13 and 17), which is 510510. Adding 2 to that gives another even number, while adding 3 would give another multiple of 3. One would similarly rule out the integers up to 18. Adding 19, however, gives 510529, which is prime. Hence 19 is a Fortunate number. The Fortunate number for ''p''''n''# is always above ''p''''n'' and all its divisors are larger than ''p''''n''. This is because ''p''''n''#, and thus ''p''''n''# + ''m'', is divisible by the prime factors of ''m'' not larger than ''p''''n''. If a composite Fortunate number does exist, it must be greater than or equal to ''p''''n+1''2. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from 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 can often be understood through the study of Complex analysis, analytical objects, such as the Riemann zeta function, that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory). One may also study real numbers in relation to rational numbers, as for instance how irrational numbers can be approximated by fractions (Diophantine approximation). Number theory is one of the oldest branches of mathematics alongside geometry. One quirk of number theory is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
37 (number)
37 (thirty-seven) is the natural number following 36 and preceding 38. In mathematics 37 is the 12th prime number, and the 3rd isolated prime without a twin prime. 37 is the first irregular prime with irregularity index of 1, where the smallest prime number with an irregularity index of 2 is the thirty-seventh prime number, 157. The smallest magic square, using only primes and 1, contains 37 as the value of its central cell: Its magic constant is 37 x 3 = 111, where 3 and 37 are the first and third base-ten unique primes (the second such prime is 11). 37 requires twenty-one steps to return to 1 in the Collatz problem, as do adjacent numbers 36 and 38. The two closest numbers to cycle through the elementary Collatz pathway are 5 and 32, whose sum is 37; also, the trajectories for 3 and 21 both require seven steps to reach 1. On the other hand, the first two integers that return 0 for the Mertens function ( 2 and 39) have a difference of 37, where the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Pages
The PrimePages is a website about prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...s originally created by Chris Caldwell at the University of Tennessee at Martin who maintained it from 1994 to 2023. The site maintains the list of the "5,000 largest known primes", selected smaller primes of special forms, and many "top twenty" lists for primes of various forms. The PrimePages has articles on primes and primality testing. It includes "The Prime Glossary" with articles on hundreds of glosses related to primes, and "Prime Curios!" with thousands of curios about specific numbers. The database started as a list of "titanic primes" (primes with at least 1000 decimal digits) by Samuel Yates in 1984. On March 11, 2023, the PrimePages moved from primes.utm.edu to t5k.or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Composite Number
A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Accordingly it is a positive integer that has at least one divisor other than 1 and itself. Every positive integer is composite, prime number, prime, or the Unit (ring theory), unit 1, so the composite numbers are exactly the numbers that are not prime and not a unit. E.g., the integer 14 is a composite number because it is the product of the two smaller integers 2 × 7 but the integers 2 and 3 are not because each can only be divided by one and itself. The composite numbers up to 150 are: :4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110, 111, 112, 114, 115, 116, 117, 118, 119, 120, 121, ... [...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 the tenth super-prime. The previous prime is 107, making them both twin primes. 109 is a centered triangular number. 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. *109 squares on an infinite chessboard that can be reached by a knight within three moves. There are 109 uniform edge-colorings to the 11 regular and semiregular (or Archimedean) tilings. 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 10 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
59 (number)
59 (fifty-nine) is the natural number following 58 (number), 58 and preceding 60 (number), 60. In mathematics Fifty-nine is the 17th prime number, and 7th super-prime. It is also a good prime, a Higgs prime, an Regular prime#Irregular primes, irregular prime, a Pillai prime, a Ramanujan prime, a Safe and Sophie Germain primes, safe prime, and a Supersingular prime (moonshine theory), supersingular prime, The next prime number is sixty-one, with which it comprises a twin prime. There are 59 stellations of the regular icosahedron. In other fields Fifty-nine is: * The number corresponding to the last minute in a given hour, and the last second in a given minute ** The "59-minute rule" is an informal rule in business, whereby (usually near a holiday) employees may be allowed to leave work early, often to beat heavy holiday traffic (the 59 minutes coming from the rule that leaving one full hour early requires the use of leave, whereas leaving 59 minutes early would not) * The number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
107 (number)
107 (one hundred ndseven) is the natural number following 106 and preceding 108. In mathematics 107 is the 28th prime number. The next prime is 109, with which it comprises a twin prime, making 107 a Chen prime. Plugged into the expression 2^p - 1, 107 yields 162259276829213363391578010288127, a Mersenne prime. 107 is itself a safe prime. It is the fourth Busy beaver number, the maximum number of steps that any Turing machine with 2 symbols and 4 states can make before eventually halting. It is the number of triangle-free graphs on 7 vertices. It is the ninth emirp, because reversing its digits gives another prime number (701) In other fields 107 is also: * The emergency telephone number in Argentina and Cape Town. * The telephone of the police in Hungary. In sports * The 107% rule, a Formula One Sporting Regulation in operation from 1996 to 2002 and 2011 onward. * The number 107 is also associated with the Timbers Army supporters group of the Portland Timbers socc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
47 (number)
47 (forty-seven) is the natural number following 46 (number), 46 and preceding 48 (number), 48. It is a prime number. It is the adopted favorite number of Pomona College, a liberal arts college in Southern California, whose alumni have added cultural references to it in numerous places, including many ''Star Trek'' episodes. Mathematics 47 is a safe prime, a Thabit number, Thabit prime, a regular prime, a cluster prime, an isolated prime, a Ramanujan prime, and a Higgs prime. 47 is also a supersingular prime (moonshine theory), supersingular prime. It is the last consecutive prime number that divides the order of at least one sporadic group. In popular culture Pomona College Other Late rapper Capital Steez was infatuated with the number 47 and what it meant spiritually. He believed the number 47 was the "perfect expression of balance in the world", representing the tension between the heart and the brain (the fourth and seventh chakras, respectively.) The number featured on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
71 (number)
71 (seventy-one) is the natural number following 70 and preceding 72. __TOC__ In mathematics 71 is the 20th prime number. Because both rearrangements of its digits (17 and 71) are prime numbers, 71 is an emirp and more generally a permutable prime. 71 is a centered heptagonal number. It is a regular prime, a Ramanujan prime, a Higgs prime, and a good prime. It is a Pillai prime, since 9!+1 is divisible by 71, but 71 is not one more than a multiple of 9. It is part of the last known pair (71, 7) of Brown numbers, since 71^=7!+1. 71 is the smallest of thirty-one discriminants of imaginary quadratic fields with class number of 7, negated (see also, Heegner number In number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from int ...s). 71 is the largest number which occurs as a prim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
67 (number)
67 (sixty-seven) is the natural number following 66 (number), 66 and preceding 68 (number), 68. It is an Parity (mathematics), odd and prime number. In mathematics 67 is: *the 19th prime number (the next is 71 (number), 71). * a Chen prime. *an irregular prime. *a lucky prime. *the sum of five consecutive primes (7 + 11 + 13 + 17 + 19). *a Heegner number. *a Pillai prime since 18! + 1 is divisible by 67, but 67 is not one more than a multiple of 18. *palindromic in quinary (2325) and senary (1516). *a super-prime. (19 is prime) *an Twin prime#Isolated prime, isolated prime. (65 and 69 are not prime) *a sexy prime with 61 and 73 References External links Integers {{Num-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
61 (number)
61 (sixty-one) is the natural number following 60 and preceding 62. In mathematics 61 is the 18th prime number, and a twin prime with 59. As a centered square number, it is the sum of two consecutive squares, 5^2 + 6^2. It is also a centered decagonal number, and a centered hexagonal number. 61 is the fourth cuban prime of the form p = \frac where x = y + 1, and the fourth Pillai prime since 8! + 1 is divisible by 61, but 61 is not one more than a multiple of 8. It is also a Keith number, as it recurs in a Fibonacci-like sequence started from its base 10 digits: 6, 1, 7, 8, 15, 23, 38, 61, ... 61 is a unique prime in base 14, since no other prime has a 6-digit period in base 14, and palindromic in bases 6 (1416) and 60 (1160). It is the sixth up/down or Euler zigzag number. 61 is the smallest ''proper prime'', a prime p which ends in the digit 1 in decimal and whose reciprocal in base-10 has a repeating sequence of length p - 1, where each digit (0, 1, ..., 9) ap ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
19 (number)
19 (nineteen) is the natural number following 18 (number), 18 and preceding 20 (number), 20. It is a prime number. Mathematics Nineteen is the eighth prime number. Number theory 19 forms a twin prime with 17 (number), 17, a cousin prime with 23 (number), 23, and a sexy prime with 13 (number), 13. 19 is the fifth Trinomial triangle#Central trinomial coefficients, central trinomial coefficient, and the maximum number of fourth powers needed to sum up to any natural number (see, Waring's problem). It is the number of Composition (combinatorics), compositions of 8 into distinct parts. 19 is the eighth strictly non-palindromic number in any Numeral system, base, following 11 (number), 11 and preceding 47 (number), 47. 19 is also the second octahedral number, after 6, and the sixth Heegner number. In the Engel expansion of pi, 19 is the seventh term following and preceding . The sum of the first terms preceding 17 (number), 17 is in equivalence with 19, where its prime Sequen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
