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Proth's theorem
In number theory, Proth's theorem is a theorem which forms the basis of a primality test for Proth numbers known as Proth's test. Proth numbers, sometimes called Proth Numbers of the First Kind, are those integers p which take the form p = k2n + 1 with an odd k where k < 2n. For Proth Numbers of the Second Kind, see related topic Riesel numbers.
The theorem states that for any Proth number (of the first kind), p, then p is prime if there exists an integer a for which Euler's criterion yields –1, that is,
In this case, p is called a Proth prime. The contrapositive is also true: if p is Proth composite, then no such a exists.
It might be noted that the presumption of k being odd does not restrict generality. So long as the condition that k < 2n is met, any factors of 2 in an even k may be factored out of k and into 2n, growing the latter while shrinking the former even further; the inequality condition remains true. Thus, k may always be reduced to an odd value.
Only one such value of a need be found for the test to deterministically confirm primality, provided that p is a Proth number. This is a practical test because, if p is prime, then any chosen a has about a 50 percent chance of working, and if p is not prime, then no chosen a will work. Furthermore, since the calculation is modulo p, only values of a smaller than p have to be considered.
If p is Proth composite, then no base a will work to bear witness of primality. If any one base a bears witness, then primality is confirmed. If none do, then compositeness is confirmed. This is because the inverse of Proth's theorem is also true:
The contrapositive of this statement is that if p is a Proth prime, such an a value is guaranteed to exist. Indeed, if p is a Proth prime then we expect roughly half of all a-values to satisfy the formula, in the general case. On the other hand, if the second condition is not met - if p is not a Proth number - then compositeness cannot be guaranteed (the inverse is not generally true for non-Proth), even if the first condition of congruence is met.
As such, we may systematically check all base values [2, p − 1] to verify compositeness (note that a = 0 and a = 1 will never work), unless and until one is found to confirm primality. This process, as it is stated, though the most straightforward and trivial, can be made more efficient.
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Proth's theorem
In number theory, Proth's theorem is a theorem which forms the basis of a primality test for Proth numbers known as Proth's test. Proth numbers, sometimes called Proth Numbers of the First Kind, are those integers p which take the form p = k2n + 1 with an odd k where k < 2n. For Proth Numbers of the Second Kind, see related topic Riesel numbers.
The theorem states that for any Proth number (of the first kind), p, then p is prime if there exists an integer a for which Euler's criterion yields –1, that is,
In this case, p is called a Proth prime. The contrapositive is also true: if p is Proth composite, then no such a exists.
It might be noted that the presumption of k being odd does not restrict generality. So long as the condition that k < 2n is met, any factors of 2 in an even k may be factored out of k and into 2n, growing the latter while shrinking the former even further; the inequality condition remains true. Thus, k may always be reduced to an odd value.
Only one such value of a need be found for the test to deterministically confirm primality, provided that p is a Proth number. This is a practical test because, if p is prime, then any chosen a has about a 50 percent chance of working, and if p is not prime, then no chosen a will work. Furthermore, since the calculation is modulo p, only values of a smaller than p have to be considered.
If p is Proth composite, then no base a will work to bear witness of primality. If any one base a bears witness, then primality is confirmed. If none do, then compositeness is confirmed. This is because the inverse of Proth's theorem is also true:
The contrapositive of this statement is that if p is a Proth prime, such an a value is guaranteed to exist. Indeed, if p is a Proth prime then we expect roughly half of all a-values to satisfy the formula, in the general case. On the other hand, if the second condition is not met - if p is not a Proth number - then compositeness cannot be guaranteed (the inverse is not generally true for non-Proth), even if the first condition of congruence is met.
As such, we may systematically check all base values [2, p − 1] to verify compositeness (note that a = 0 and a = 1 will never work), unless and until one is found to confirm primality. This process, as it is stated, though the most straightforward and trivial, can be made more efficient.