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Often a second, additional meaning is intended by using the word prime, namely that any object can be, essentially uniquely, decomposed into its prime components. For example, in knot theory , a prime knot is a knot that is indecomposable in the sense that it cannot be written as the connected sum of two nontrivial knots.
All prime numbers from 31 to 6,469,693,189 for free download. Lists of Primes at the Prime Pages. The Nth Prime Page Nth prime through n=10^12, pi(x) through x=3*10^13, Random primes in same range. Interface to a list of the first 98 million primes (primes less than 2,000,000,000) Weisstein, Eric W. "Prime Number Sequences". MathWorld.
For example: "All humans are mortal, and Socrates is a human. ∴ Socrates is mortal." ∵ Abbreviation of "because" or "since". Placed between two assertions, it means that the first one is implied by the second one. For example: "11 is prime ∵ it has no positive integer factors other than itself and one." ∋ 1. Abbreviation of "such that".
In mathematics, the prime is generally used to generate more variable names for similar things without resorting to subscripts, with x ′ generally meaning something related to (or derived from) x. For example, if a point is represented by the Cartesian coordinates ( x , y ) , then that point rotated, translated or reflected might be ...
An example of an active area of research in algebraic number theory is Iwasawa theory. The Langlands program, one of the main current large-scale research plans in mathematics, is sometimes described as an attempt to generalise class field theory to non-abelian extensions of number fields.
For example, "almost all prime numbers are odd". There is a more complicated meaning for integers as well, discussed in the main article. Finally, this term is sometimes used synonymously with generic, below. arbitrarily large Notions which arise mostly in the context of limits, referring to the recurrence of a phenomenon as the limit is approached
As another example, φ(1) = 1 since for n = 1 the only integer in the range from 1 to n is 1 itself, and gcd(1, 1) = 1. Euler's totient function is a multiplicative function, meaning that if two numbers m and n are relatively prime, then φ(mn) = φ(m)φ(n).
For example, 123.456 represents 123456 / 1000 , or, in words, one hundred, two tens, three ones, four tenths, five hundredths, and six thousandths. A real number can be expressed by a finite number of decimal digits only if it is rational and its fractional part has a denominator whose prime factors are 2 or 5 or both, because these are ...