There aren't enough small numbers to meet the many demands made of them.
In other words, any given small number appears in far more contexts than may seem reasonable, leading to many apparently surprising coincidences in mathematics, simply because small numbers appear so often and yet are so few. Earlier (1980) this "law" was reported by Martin Gardner.[2] Guy's subsequent 1988 paper of the same title gives numerous examples in support of this thesis. (This paper earned him the MAA Lester R. Ford Award.)
Second strong law of small numbers
Guy gives Moser's circle problem as an example. The number of points (n),chords (c) and regions (rG). The first five terms for the number of regions follow a simple sequence, broken by the sixth term.
Guy also formulated a Second strong law of small numbers:
When two numbers look equal, it ain't necessarily so![3]
Guy explains this latter law by the way of examples: he cites numerous sequences for which observing the first few members may lead to a wrong guess about the generating formula or law for the sequence. Many of the examples are the observations of other mathematicians.[3]
One example Guy gives is the conjecture that is prime—in fact, a Mersenne prime—when is prime; but this conjecture, while true for = 2, 3, 5 and 7, fails for = 11 (and for many other values).
Another relates to the Prime number race: primes congruent to 3 modulo 4 appear to be more numerous than those congruent to 1; however this is false, and first ceases being true at 26861.
Carnahan, Scott (2007-10-27). "Small finite sets". Secret Blogging Seminar, notes on a talk by Jean-Pierre Serre on properties of small finite sets.CS1 maint: postscript (link)
Amos Tversky; Daniel Kahneman (August 1971). "Belief in the law of small numbers". Psychological Bulletin. 76 (2): 105–110. CiteSeerX10.1.1.592.3838. doi:10.1037/h0031322. people have erroneous intuitions about the laws of chance. In particular, they regard a sample randomly drawn from a population as highly representative, I.e., similar to the population in all essential characteristics.
A prime number is a natural number greater than 1 that is not a 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 because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself. However, 4 is composite because it is a product in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order.
The Ulam spiral or prime spiral is a graphical depiction of the set of prime numbers, devised by mathematician Stanisław Ulam in 1963 and popularized in Martin Gardner's Mathematical Games column in Scientific American a short time later. It is constructed by writing the positive integers in a square spiral and specially marking the prime numbers.
In number theory, a prime number p is a Sophie Germain prime if 2p + 1 is also prime. The number 2p + 1 associated with a Sophie Germain prime is called a safe prime. For example, 11 is a Sophie Germain prime and 2 × 11 + 1 = 23 is its associated safe prime. Sophie Germain primes are named after French mathematician Sophie Germain, who used them in her investigations of Fermat's Last Theorem. Sophie Germain primes and safe primes have applications in public key cryptography and primality testing. It has been conjectured that there are infinitely many Sophie Germain primes, but this remains unproven.
In mathematics, parity is the property of an integer of whether it is even or odd. An integer's parity is even if it is divisible by two with no remainders left and its parity is odd if its remainder is 1. For example, -4, 0, 82, and 178 are even because there is no remainder when dividing it by 2. By contrast, -3, 5, 7, 21 are odd numbers as they leave a remainder of 1 when divided by 2.
In mathematics, the Langlands program is a web of far-reaching and influential conjectures about connections between number theory and geometry. Proposed by Robert Langlands, it seeks to relate Galois groups in algebraic number theory to automorphic forms and representation theory of algebraic groups over local fields and adeles. Widely seen as the single biggest project in modern mathematical research, the Langlands program has been described by Edward Frenkel as "a kind of grand unified theory of mathematics."
The abc conjecture is a conjecture in number theory, first proposed by Joseph Oesterlé (1988) and David Masser (1985). It is stated in terms of three positive integers, a, b and c that are relatively prime and satisfy a + b = c. If d denotes the product of the distinct prime factors of abc, the conjecture essentially states that d is usually not much smaller than c. In other words: if a and b are composed from large powers of primes, then c is usually not divisible by large powers of primes. A number of famous conjectures and theorems in number theory would follow immediately from the abc conjecture or its versions. Goldfeld (1996) described the abc conjecture as "the most important unsolved problem in Diophantine analysis".
In mathematics, a Cullen number is a member of the integer sequence . Cullen numbers were first studied by James Cullen in 1905. The numbers are special cases of Proth numbers.
In number theory, a Wieferich prime is a prime number p such that p2 divides 2p − 1 − 1, therefore connecting these primes with Fermat's little theorem, which states that every odd prime p divides 2p − 1 − 1. Wieferich primes were first described by Arthur Wieferich in 1909 in works pertaining to Fermat's last theorem, at which time both of Fermat's theorems were already well known to mathematicians.
An Egyptian fraction is a finite sum of distinct unit fractions, such as
In mathematics, a constructive proof is a method of proof that demonstrates the existence of a mathematical object by creating or providing a method for creating the object. This is in contrast to a non-constructive proof, which proves the existence of a particular kind of object without providing an example. For avoiding confusion with the stronger concept that follows, such a constructive proof is sometimes called an effective proof.
In mathematics, Schinzel's hypothesis H is one of the most famous open problems in the topic of number theory. It is a very broad generalisation of widely open conjectures such as the twin prime conjecture. The hypothesis is named after Andrzej Schinzel.
Richard Kenneth Guy was a British mathematician. He was a professor in the Department of Mathematics at the University of Calgary. He is known for his work in number theory, geometry, recreational mathematics, combinatorics, and graph theory. He is best known for co-authorship of Winning Ways for your Mathematical Plays and authorship of Unsolved Problems in Number Theory. He published more than 300 scholarly articles. Guy proposed the partially tongue-in-cheek "Strong Law of Small Numbers", which says there are not enough small integers available for the many tasks assigned to them – thus explaining many coincidences and patterns found among numerous cultures. For this paper he received the MAA Lester R. Ford Award.
Gilbreath's conjecture is a conjecture in number theory regarding the sequences generated by applying the forward difference operator to consecutive prime numbers and leaving the results unsigned, and then repeating this process on consecutive terms in the resulting sequence, and so forth. The statement is named after Norman L. Gilbreath who, in 1958, presented it to the mathematical community after observing the pattern by chance while doing arithmetic on a napkin. In 1878, eighty years before Gilbreath's discovery, François Proth had, however, published the same observations along with an attempted proof, which was later shown to be false.
In number theory, the Bateman–Horn conjecture is a statement concerning the frequency of prime numbers among the values of a system of polynomials, named after mathematicians Paul T. Bateman and Roger A. Horn who proposed it in 1962. It provides a vast generalization of such conjectures as the Hardy and Littlewood conjecture on the density of twin primes or their conjecture on primes of the form n2 + 1; it is also a strengthening of Schinzel's hypothesis H.
In mathematics, a Thue equation is a Diophantine equation of the form
In mathematics, a congruent number is a positive integer that is the area of a right triangle with three rational number sides. A more general definition includes all positive rational numbers with this property.
In number theory, friendly numbers are two or more natural numbers with a common abundancy index, the ratio between the sum of divisors of a number and the number itself. Two numbers with the same "abundancy" form a friendly pair; n numbers with the same "abundancy" form a friendly n-tuple.
In number theory, Sylvester's sequence is an integer sequence in which each term of the sequence is the product of the previous terms, plus one. The first few terms of the sequence are
At the 1912 International Congress of Mathematicians, Edmund Landau listed four basic problems about prime numbers. These problems were characterised in his speech as "unattackable at the present state of mathematics" and are now known as Landau's problems. They are as follows:
Goldbach's conjecture: Can every even integer greater than 2 be written as the sum of two primes?
Twin prime conjecture: Are there infinitely many primes p such that p + 2 is prime?
Legendre's conjecture: Does there always exist at least one prime between consecutive perfect squares?
Are there infinitely many primes p such that p − 1 is a perfect square? In other words: Are there infinitely many primes of the form n2 + 1?
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