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25 (number)
25 (twenty-five) is the natural number following 24 and preceding 26. In mathematics It is a square number, being 52 = 5 × 5, and hence the third non-unitary square prime of the form ''p''2. It is one of two two-digit numbers whose square and higher powers of the number also ends in the same last two digits, e.g., 252 = 625; the other is 76. 25 has an even aliquot sum of 6, which is itself the first even and perfect number root of an aliquot sequence; not ending in ( 1 and 0). It is the smallest square that is also a sum of two (non-zero) squares: 25 = 32 + 42. Hence, it often appears in illustrations of the Pythagorean theorem. 25 is the sum of the five consecutive single-digit odd natural numbers 1, 3, 5, 7, and 9. 25 is a centered octagonal number, a centered square number, a centered octahedral number, and an automorphic number. 25 percent (%) is equal to . It is the smallest decimal Friedman number as it can be expressed by its own digits: 52. It ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Number
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive integers Some authors acknowledge both definitions whenever convenient. Sometimes, the whole numbers are the natural numbers as well as zero. In other cases, the ''whole numbers'' refer to all of the integers, including negative integers. The counting numbers are another term for the natural numbers, particularly in primary education, and are ambiguous as well although typically start at 1. The natural numbers are used for counting things, like "there are ''six'' coins on the table", in which case they are called ''cardinal numbers''. They are also used to put things in order, like "this is the ''third'' largest city in the country", which are called ''ordinal numbers''. Natural numbers are also used as labels, like Number (sports), jersey ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cullen Number
In mathematics, a Cullen number is a member of the integer sequence C_n = n \cdot 2^n + 1 (where n is a natural number). Cullen numbers were first studied by James Cullen in 1905. The numbers are special cases of Proth numbers. Properties In 1976 Christopher Hooley showed that the natural density of positive integers n \leq x for which ''C''''n'' is a prime is of the order ''o''(''x'') for x \to \infty. In that sense, almost all Cullen numbers are composite. Hooley's proof was reworked by Hiromi Suyama to show that it works for any sequence of numbers ''n''·2''n'' + ''a'' + ''b'' where ''a'' and ''b'' are integers, and in particular also for Woodall numbers. The only known Cullen primes are those for ''n'' equal to: : 1, 141, 4713, 5795, 6611, 18496, 32292, 32469, 59656, 90825, 262419, 361275, 481899, 1354828, 6328548, 6679881 . Still, it is conjectured that there are infinitely many Cullen primes. A Cullen number ''C''''n'' is divisible by ''p'' = 2''n'' ... [...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] |
31 (number)
31 (thirty-one) is the natural number following thirty, 30 and preceding 32 (number), 32. It is a prime number. Mathematics 31 is the 11th prime number. It is a superprime and a Self number#Self primes, self prime (after 3, 5, and 7), as no integer added up to its base 10 digits results in 31. It is the third Mersenne prime of the form 2''n'' − 1, and the eighth Mersenne prime ''exponent'', in-turn yielding the maximum positive value for a 32-bit Integer (computer science), signed binary integer in computing: 2,147,483,647. After 3, it is the second Mersenne prime not to be a double Mersenne prime, while the 31st prime number (127 (number), 127) is the second double Mersenne prime, following 7. On the other hand, the thirty-first triangular number is the perfect number 496 (number), 496, of the form 2(5 − 1)(25 − 1) by the Euclid-Euler theorem. 31 is also a ''primorial prime'' like its twin prime (29 (number), 29), as well as both a lucky prime and a happy number like its d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
29 (number)
29 (twenty-nine) is the natural number following 28 and preceding 30. It is a prime number. 29 is the number of days February has on a leap year. Mathematics 29 is the tenth prime number. Integer properties 29 is the fifth primorial prime, like its twin prime 31. 29 is the smallest positive whole number that cannot be made from the numbers \, using each digit exactly once and using only addition, subtraction, multiplication, and division. None of the first twenty-nine natural numbers have more than two different prime factors (in other words, this is the longest such consecutive sequence; the first sphenic number or triprime, 30 is the product of the first three primes 2, 3, and 5). 29 is also, * the sum of three consecutive squares, 22 + 32 + 42. * the sixth Sophie Germain prime. * a Lucas prime, a Pell prime, and a tetranacci number. * an Eisenstein prime with no imaginary part and real part of the form 3n − 1. * a Markov number, appearing in the solution ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
23 (number)
23 (twenty-three) is the natural number following 22 and preceding 24. It is a prime number. 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 and 29; while also being the largest member of the first prime sextuplet ( 7, 11, 13, 17, 19, 23). Twenty-three is also the next to last member of the first Cunningham chain of the first kind ( 2, 5, 11, 23, 47), and the sum of the prime factors of the second set of consecutive discrete semiprimes, ( 21, 22). 23 is the smallest odd prime to be a highly cototient number, as the solution to x-\phi(x) for the integers 95, 119, 143, and 529. * 23 is the second Smarandache–Wellin prime in base ten, as it is the concatenation of the decimal representations of the first two primes (2 and 3) and is itself also prime, and a happy number. * The sum of the first nine primes up to 23 is a square: 2 + ... [...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] |
17 (number)
17 (seventeen) is the natural number following 16 (number), 16 and preceding 18 (number), 18. It is a prime number. 17 was described at MIT as "the least random number", according to the Jargon File. This is supposedly because, in a study where respondents were asked to choose a random number from 1 to 20, 17 was the most common choice. This study has been repeated a number of times. Mathematics 17 is a Leyland number and Leyland number#Leyland primes, Leyland prime, using 2 & 3 (23 + 32) and using 4 and 5, using 3 & 4 (34 - 43). 17 is a Fermat prime. 17 is one of six lucky numbers of Euler. Since seventeen is a Fermat prime, regular heptadecagons can be constructible polygon, constructed with a compass and unmarked ruler. This was proven by Carl Friedrich Gauss and ultimately led him to choose mathematics over philology for his studies. The minimum possible number of givens for a sudoku puzzle with a unique solution is 17. Geometric properties Two-dimensions *There are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
13 (number)
13 (thirteen) is the natural number following 12 (number), 12 and preceding 14 (number), 14. Folklore surrounding the number 13 appears in many cultures around the world: one theory is that this is due to the cultures employing lunar-solar calendars (there are approximately 12.41 lunations per solar year, and hence 12 "true months" plus a smaller, and often portentous, thirteenth month). This can be witnessed, for example, in the "Twelve Days of Christmas" of Western European tradition. In mathematics The number 13 is a prime number, happy number and a lucky number. It is a twin prime with 11 (number), 11, as well as a cousin prime with 17 (number), 17. It is the second of only 3 Wilson prime, Wilson primes: 5, 13, and 563 (number), 563. A 13-sided regular polygon is called a tridecagon. List of basic calculations In languages Grammar * In all Germanic languages, 13 is the first Compound (linguistics), compound number; the numbers 11 and 12 have their own names. * The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
11 (number)
11 (eleven) is the natural number following 10 and preceding 12 (number), 12. It is the smallest number whose name has three syllables. Name "Eleven" derives from the Old English ', which is first attested in Bede's late 9th-century ''Ecclesiastical History of the English People''. It has cognates in every Germanic language (for example, German ), whose Proto-Germanic language, Proto-Germanic ancestor has been linguistic reconstruction, reconstructed as , from the prefix (adjectival "1 (number), one") and suffix , of uncertain meaning. It is sometimes compared with the Lithuanian language, Lithuanian ', though ' is used as the suffix for all numbers from 11 to 19. The Old English form has closer cognates in Old Frisian, Old Saxon, Saxon, and Old Norse, Norse, whose ancestor has been reconstructed as . This was formerly thought to be derived from Proto-Germanic ("10 (number), ten"); it is now sometimes connected with or ("left; remaining"), with the implicit meaning that "one is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shapiro Inequality
In mathematics, the Shapiro inequality is an inequality proposed by Harold S. Shapiro in 1954. Statement of the inequality Suppose is a natural number In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ... and are positive numbers and: * is even and less than or equal to , or * is odd and less than or equal to . Then the Shapiro inequality states that :\sum_^n \frac \geq \frac, where and . The special case with is Nesbitt's inequality. For greater values of the inequality does not hold, and the strict lower bound is with . The initial proofs of the inequality in the pivotal cases and rely on numerical computations. In 2002, P.J. Bushell and J.B. McLeod published an analytical proof for . The value of was determined in 1971 by Vladimir Drinfeld. Specifi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aliquot Sequence
In mathematics, an aliquot sequence is a sequence of positive integers in which each term is the sum of the proper divisors of the previous term. If the sequence reaches the number 1, it ends, since the sum of the proper divisors of 1 is 0. Definition and overview The aliquot sequence starting with a positive integer can be defined formally in terms of the sum-of-divisors function or the aliquot sum function in the following way: \begin s_0 &= k \\ pts_n &= s(s_) = \sigma_1(s_) - s_ \quad \text \quad s_ > 0 \\ pts_n &= 0 \quad \text \quad s_ = 0 \\ pts(0) &= \text \end If the condition is added, then the terms after 0 are all 0, and all aliquot sequences would be infinite, and we can conjecture that all aliquot sequences are convergent, the limit of these sequences are usually 0 or 6. For example, the aliquot sequence of 10 is because: \begin \sigma_1(10) -10 &= 5 + 2 + 1 = 8, \\ pt\sigma_1(8) - 8 &= 4 + 2 + 1 = 7, \\ pt\sigma_1(7) - 7 &= 1, \\ pt\sigma_1(1) - 1 &= 0. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
