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Perfect Number
In number theory, a perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself (also known as its aliquot sum). Equivalently, a perfect number is a number that is half the sum of all of its positive divisors (including itself) i.e. σ1(n) = 2n. This definition is ancient, appearing as early as Euclid's Elements (VII.22) where it is called τέλειος ἀριθμός (perfect, ideal, or complete number). Euclid
Euclid
also proved a formation rule (IX.36) whereby q ( q + 1 ) / 2 displaystyle q(q+1)/2 is an even perfect number whenever q displaystyle q is a prime of the form 2 p − 1 displaystyle 2^ p -1 for prime p displaystyle p —what is now called a Mersenne prime
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Deficient Number
In number theory, a deficient or deficient number is a number n for which the sum of divisors σ(n)<2n, or, equivalently, the sum of proper divisors (or aliquot sum) s(n)<n. The value 2n − σ(n) (or n − s(n)) is called the number's deficiency.Contents1 Examples 2 Properties 3 Related concepts 4 See also 5 References 6 External linksExamples[edit] The first few deficient numbers are:1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50, ... (sequence A005100 in the OEIS)As an example, consider the number 21. Its proper divisors are 1, 3 and 7, and their sum is 11. Because 11 is less than 21, the number 21 is deficient
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Jacques Lefèvre D'Étaples
Jacques Lefèvre d' Étaples
Étaples
or Jacobus Faber Stapulensis[1] (c. 1455 – 1536) was a French theologian and humanist. He was a precursor of the Protestant
Protestant
movement in France. The "d'Étaples" was not part of his name as such, but used to distinguish him from Jacques Lefèvre of Deventer, a less significant contemporary, a friend and correspondent of Erasmus. Both are also sometimes called by the German version of their name, Jacob/Jakob Faber
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Bijection
In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set. There are no unpaired elements. In mathematical terms, a bijective function f: X → Y is a one-to-one (injective) and onto (surjective) mapping of a set X to a set Y. A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are finite sets, then the existence of a bijection means they have the same number of elements
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GIMPS
The Great Internet Mersenne Prime Search (GIMPS) is a collaborative project of volunteers who use freely available software to search for Mersenne prime
Mersenne prime
numbers. The GIMPS project was founded by George Woltman, who also wrote the software Prime95
Prime95
and MPrime
MPrime
for the project. Scott Kurowski wrote the PrimeNet Internet server
Internet server
that supports the research to demonstrate Entropia-distributed computing software, a company he founded in 1997. GIMPS is registered as Mersenne Research, Inc. Kurowski is Executive Vice President and board director of Mersenne Research Inc
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Infinite Set
In set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable. Some examples are:the set of all integers, ..., -1, 0, 1, 2, ... , is a countably infinite set; and the set of all real numbers is an uncountably infinite set.Contents1 Properties 2 See also 3 References 4 External linksProperties[edit] The set of natural numbers (whose existence is postulated by the axiom of infinity) is infinite. It is the only set that is directly required by the axioms to be infinite
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Triangular Number
A triangular number or triangle number counts objects arranged in an equilateral triangle, as in the diagram on the right. The nth triangular number is the number of dots in the triangular arrangement with n dots on a side, and is equal to the sum of the n natural numbers from 1 to n
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Centered Nonagonal Number
A centered nonagonal number is a centered figurate number that represents a nonagon with a dot in the center and all other dots surrounding the center dot in successive nonagonal layers. The centered nonagonal number for n is given by the formula[1] N c ( n ) = ( 3 n − 2 ) ( 3 n − 1 ) 2 . displaystyle Nc(n)= frac (3n-2)(3n-1) 2
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Digital Root
The digital root (also repeated digital sum) of a non-negative integer is the (single digit) value obtained by an iterative process of summing digits, on each iteration using the result from the previous iteration to compute a digit sum. The process continues until a single-digit number is reached. For example, the digital root of 65,536 is 7, because 6 + 5 + 5 + 3 + 6 = 25 and 2 + 5 = 7. Digital roots can be calculated with congruences in modular arithmetic rather than by adding up all the digits, a procedure that can save time in the case of very large numbers. Digital roots can be used as a sort of checksum, to check that a sum has been performed correctly. If it has, then the digital root of the sum of the given numbers will equal the digital root of the sum of the digital roots of the given numbers
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Pernicious Number
In number theory, a pernicious number is a positive integer where the Hamming weight (or digit sum) of its binary representation is prime.Contents1 Examples 2 Properties 3 Related numbers 4 References 5 External linksExamples[edit] The first pernicious number is 3, since 3 = 112 and 1 + 1 = 2, which is a prime. The next pernicious number is 5, since 5 = 1012, followed by 6, 7 and 9 (sequence A052294 in the OEIS). Properties[edit]No power of two is a pernicious number. This is trivially true, because powers of two in binary form are represented as a one followed by zeros. So each power of two has a Hamming weight of one, and one is not considered to be a prime. Every number of the form 2n + 1 with n > 0, including every Fermat number, is a pernicious number. This is because the sum of the digits in binary form is 2, which is a prime number. Every even perfect number is a pernicious number
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Practical Number
In number theory, a practical number or panarithmic number[1] is a positive integer n such that all smaller positive integers can be represented as sums of distinct divisors of n. For example, 12 is a practical number because all the numbers from 1 to 11 can be expressed as sums of its divisors 1, 2, 3, 4, and 6: as well as these divisors themselves, we have 5 = 3 + 2, 7 = 6 + 1, 8 = 6 + 2, 9 = 6 + 3, 10 = 6 + 3 + 1, and 11 = 6 + 3 + 2. The sequence of practical numbers (sequence A005153 in the OEIS) begins1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 48, 54, 56, 60, 64, 66, 72, 78, 80, 84, 88, 90, 96, 100, 104, 108, 112, 120, 126, 128, 132, 140, 144, 150....Practical numbers were used by Fibonacci
Fibonacci
in his Liber Abaci
Liber Abaci
(1202) in connection with the problem of representing rational numbers as Egyptian fractions
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Carl Pomerance
Carl Bernard Pomerance (born in 1944 in Joplin, Missouri) is an American number theorist. He attended college at Brown University
Brown University
and later received his Ph.D.
Ph.D.
from Harvard University
Harvard University
in 1972 with a dissertation proving that any odd perfect number has at least seven distinct prime factors.[1] He joined the faculty at the University of Georgia, becoming full professor in 1982. He subsequently worked at Lucent Technologies
Lucent Technologies
for a number of years, and then became a distinguished Professor at Dartmouth College.Contents1 Contributions 2 Awards and honors 3 See also 4 References 5 External linksContributions[edit] He has over 120 publications, including co-authorship with Richard Crandall of Prime numbers: a computational perspective (Springer-Verlag, first edition 2001, second edition 2005[2])
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Ibn Al-Haytham
Hasan Ibn al-Haytham
Ibn al-Haytham
(Latinized Alhazen[8] /ˌælˈhɑːzən/; full name Abū ʿAlī al-Ḥasan ibn al-Ḥasan ibn al-Haytham أبو علي، الحسن بن الحسن بن الهيثم; c. 965 – c. 1040) was an Arab[9][10][11][12][13] mathematician, astronomer, and physicist of the Islamic Golden Age.[14] He made significant contributions to the principles of optics and visual perception in particular, his most influential work being his Kitāb al-Manāẓir (كتاب المناظر, "Book of Optics"), written during 1011–1021, which survived in the
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Heuristic Argument
A heuristic argument is an argument that reasons from the value of a method or principle that has been shown by experimental (especially trial-and-error) investigation to be a useful aid in learning, discovery and problem-solving. A widely used and important example of a heuristic argument is Occam's Razor. It is a speculative, non-rigorous argument, that relies on an analogy or on intuition, that allows one to achieve a result or approximation to be checked later with more rigor; otherwise the results are of doubt. It is used as a hypothesis or conjecture in an investigation. It can also be used as a mnemonic.[1] See also[edit]Rule of thumb Probabilistic method Empirical relationshipReferences[edit]^ Brodsky, Stanley J.; Ellis, John; Karliner, Marek (1988). "Chiral symmetry and the spin of the proton" (PDF). Physics Letters B. 206 (2): 309–315. doi:10.1016/0370-2693(88)91511-0. This philosophy-related article is a stub
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Modular Arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" upon reaching a certain value—the modulus (plural moduli). The modern approach to modular arithmetic was developed by Carl Friedrich Gauss
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. Usual addition would suggest that the later time should be 7 + 8 = 15, but this is not the answer because clock time "wraps around" every 12 hours
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