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The next_prime function relies on nextprime from PARI, which finds the next pseudoprime number as defined on the page https://pari.math.u-bordeaux.fr/dochtml/html/Arithmetic_functions.html#se:ispseudoprime

A prime number is pseudoprime and "most" non-prime numbers are not pseudoprime. The benefit of such function is, as you noticed, checking that a number is pseudoprime is much faster than checking that a number is prime.

The next_prime function relies on nextprime from PARI, which finds the next pseudoprime number as defined on the page https://pari.math.u-bordeaux.fr/dochtml/html/Arithmetic_functions.html#se:ispseudoprime

A prime number is pseudoprime and "most" non-prime numbers are not pseudoprime. The benefit of such function is, as you noticed, checking that a number is pseudoprime is much faster than checking that a number is prime.

The next_prime function relies on nextprime from PARI, which finds the next pseudoprime number as defined on the page https://pari.math.u-bordeaux.fr/dochtml/html/Arithmetic_functions.html#se:ispseudoprime

A prime number is pseudoprime and "most" non-prime numbers are not pseudoprime. The benefit of such function is, as you noticed, checking that a number is pseudoprime is much faster than checking that a number is prime.

According to the doc, next_prime proves that the returned number is actually a prime number by default, which is not the case for next_probable_prime, though no example of a probable prime that is not a genuine prime was found yet according to the doc (and no such number $\leq 2^64$ 2^{64}$ exist).

The next_prime function relies on nextprime from PARI, which finds the next pseudoprime number as defined on the page https://pari.math.u-bordeaux.fr/dochtml/html/Arithmetic_functions.html#se:ispseudoprime

A prime number is pseudoprime and "most" non-prime numbers are not pseudoprime. The benefit of such function is, as you noticed, checking that a number is pseudoprime is much faster than checking that a number is prime.

According to the doc, next_prime proves that the returned number is actually a prime number by default, which is not the case for next_probable_prime, though no example of a probable prime that is not a genuine prime was found yet according (according to the doc (and doc), and no such number $\leq 2^{64}$ exist).exist.

The next_prime function relies on nextprime from PARI, which finds the next pseudoprime number as defined on the page https://pari.math.u-bordeaux.fr/dochtml/html/Arithmetic_functions.html#se:ispseudoprime

A prime number is pseudoprime and "most" non-prime numbers are not pseudoprime. The benefit of such function is, as you noticed, checking that a number is pseudoprime is much faster than checking that a number is prime.

According to the doc, by default next_prime proves that the returned number is actually a prime number by default, number, which is not the case for next_probable_prime, though no example of a probable prime that is not a genuine prime was found yet (according to the doc), and no such number $\leq 2^{64}$ exist.

The next_prime function relies on nextprime from PARI, which finds the next pseudoprime number as defined on the page https://pari.math.u-bordeaux.fr/dochtml/html/Arithmetic_functions.html#se:ispseudoprime

A prime number is pseudoprime and "most" non-prime numbers are not pseudoprime. The benefit of such function is, as you noticed, checking that a number is pseudoprime is much faster than checking that a number is prime.

According to the Sage doc, by default next_prime proves that the returned number is actually a prime number, which is not the case for next_probable_prime, though no example of a probable prime that is not a genuine prime was found yet (according to the doc), and no such number $\leq 2^{64}$ exist.

The next_prime function relies on nextprime from PARI, which finds the next pseudoprime number as defined on the page https://pari.math.u-bordeaux.fr/dochtml/html/Arithmetic_functions.html#se:ispseudoprime

A prime number is pseudoprime and "most" non-prime numbers are not pseudoprime. The benefit of such function is, as you noticed, checking that a number is pseudoprime is much faster than checking that a number is prime.

According to the Sage doc, by default next_prime proves that the returned number is actually a prime number, which is not the case for next_probable_prime, though no example of a probable prime that is not a genuine prime was found yet (according to the doc), and no such number $\leq 2^{64}$ exist.

The next_prime function relies on nextprime from PARI, which finds the next pseudoprime number as defined on the page https://pari.math.u-bordeaux.fr/dochtml/html/Arithmetic_functions.html#se:ispseudoprime

A prime number is pseudoprime and "most" non-prime numbers are not pseudoprime. The benefit of such function is, as you noticed, checking that a number is pseudoprime is much faster than checking that a number is prime.

According to the Sage doc, by default next_prime proves that the returned number is actually a prime number, which is not the case for next_probable_prime, though no example of a probable prime that is not a genuine prime was found yet (according to the doc), and no such number $\leq 2^{64}$ exist.