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A cluster prime is a prime p such that every even natural number k ≤ p − 3 is the difference of two primes not exceeding p. 3 , 5 , 7 , 11 , 13 , 17 , 19 , 23 , ... ( OEIS : A038134 )
In the second step, they were divided by 3. The final result, 4 / 3 , is an irreducible fraction because 4 and 3 have no common factors other than 1. The original fraction could have also been reduced in a single step by using the greatest common divisor of 90 and 120, which is 30. As 120 ÷ 30 = 4, and 90 ÷ 30 = 3, one gets
The numbers 8 and 9 are coprime, despite the fact that neither—considered individually—is a prime number, since 1 is their only common divisor. On the other hand, 6 and 9 are not coprime, because they are both divisible by 3. The numerator and denominator of a reduced fraction are coprime, by definition.
Similarly, when written in the usual decimal system, all prime numbers larger than 5 end in 1, 3, 7, or 9. The numbers that end with other digits are all composite: decimal numbers that end in 0, 2, 4, 6, or 8 are even, and decimal numbers that end in 0 or 5 are divisible by 5. [11]
Because the set of primes is a computably enumerable set, by Matiyasevich's theorem, it can be obtained from a system of Diophantine equations. Jones et al. (1976) found an explicit set of 14 Diophantine equations in 26 variables, such that a given number k + 2 is prime if and only if that system has a solution in nonnegative integers: [7]
After much thinking, Handford came up with the idea of "Wally", a world traveller and time travel aficionado who always dresses in red and white. [4] Sometimes it would take him up to eight weeks to draw a two-page sketch of the elusive "Wally" and the characters surrounding him. [5] The first Where's Wally? book was published on 21 September ...
For example, 15 is a composite number because 15 = 3 · 5, but 7 is a prime number because it cannot be decomposed in this way. If one of the factors is composite, it can in turn be written as a product of smaller factors, for example 60 = 3 · 20 = 3 · (5 · 4) .
If p is composite, the probability that the test declares it a strong probable prime anyway is at most 1 ⁄ 4, in which case p is a strong pseudoprime, and a is a strong liar. Therefore after k non-conclusive random tests, the probability that p is composite is at most 4 − k , and may thus be made as low as desired by increasing k .