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The Miller–Rabin primality test and Solovay–Strassen primality test are more sophisticated variants, which detect all composites (once again, this means: for every composite number n, at least 3/4 (Miller–Rabin) or 1/2 (Solovay–Strassen) of numbers a are witnesses of compositeness of n). These are also compositeness tests.
A number n that has more divisors than any x < n is a highly composite number (though the first two such numbers are 1 and 2). Composite numbers have also been called "rectangular numbers", but that name can also refer to the pronic numbers, numbers that are the product of two consecutive integers. Yet another way to classify composite numbers ...
A strong Fibonacci pseudoprime is a composite number n for which congruence holds for Q = −1 and all P. [17] It follows [17]: 460 that an odd composite integer n is a strong Fibonacci pseudoprime if and only if: n is a Carmichael number; 2(p + 1) | (n − 1) or 2(p + 1) | (n − p) for every prime p dividing n.
For a fixed base a, it is unusual for a composite number to be a probable prime (that is, a pseudoprime) to that base. For example, up to 25 × 10 9, there are 11,408,012,595 odd composite numbers, but only 21,853 pseudoprimes base 2. [1]: 1005 The number of odd primes in the same interval is 1,091,987,404.
For a positive integer a, if a composite integer x divides a x−1 − 1, then x is called a Fermat pseudoprime to base a. [1]: Def. 3.32 In other words, a composite integer is a Fermat pseudoprime to base a if it successfully passes the Fermat primality test for the base a. [2]
So if it is unknown whether a number n is prime or composite, we can pick a random number a, calculate the Jacobi symbol ( a / n ) and compare it with Euler's formula; if they differ modulo n, then n is composite; if they have the same residue modulo n for many different values of a, then n is "probably prime".
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Highly composite numbers greater than 6 are also abundant numbers. One need only look at the three largest proper divisors of a particular highly composite number to ascertain this fact. It is false that all highly composite numbers are also Harshad numbers in base 10. The first highly composite number that is not a Harshad number is ...