Equivalence of recursive specifications in process algebra

We define an equivalence relation on recursive specifications in process algebra that is model-independent and does not involve an explicit notion of solution. Then we extend this equivalence to the specification language $mu$CRL

Computer verification of the Ankeny-Artin-Chowla conjecture for all primes less than 100,000,000,000100,000,000,000

An error in the program for verifying the Ankeny-Artin-Chowla (AAC) conjecture is reported. As a result, in the case of primes p which are ≡ 5 mod 8, the AAC conjecture has been verified using a different multiple of the regulator of the quadratic field ℚ(Formula Presented) than was meant. However, since any multiple of this regulator is suitable for this purpose, provided that it is smaller than 8p, the main result that the AAC conjecture is true for all the primes ≡ 1 mod 4 which are 1011, remains valid. As an addition, we have verified the AAC conjecture for all the primes ≡ 1 mod 4 between 1011 and 2 × 1011, with the corrected program

Computer verification of the Ankeny-Artin-Chowla conjecture for all primes less than 100 000 000 000

Let $p$ be a prime congruent to 1 modulo~4 and let $t, u$ be rational integers such that (t+usqrt{p,)/2 is the fundamental unit of the real quadratic field {mathbb Q(sqrt{p,). The Ankeny-Artin-Chowla Conjecture (AACC) asserts that $p$ will not divide $u$. This is equivalent to the assertion that $p$ will not divide B_{(p-1)/2, where B_{n denotes the n^{th Bernoulli number. Although first published in 1952, this conjecture still remains unproved today. Indeed, it appears to be most difficult to prove. Even testing the conjecture can be quite challenging because of the size of the numbers $t, u$; for example, when $p = 40,094,470,441$, then both $t$ and $u$ exceed 10^{330000. In 1988 the AAC conjecture was verified by computer for all p < 10^{9. In this paper we describe a new technique for testing the AAC conjecture and we provide some results of a computer run of the method for all primes $p$ up to 10^{11

Corrigenda and addition to \computer verification of the Ankeny-Artin-Chowla conjecture for all primes less than 100 000 000 000"

An error in the program for verifying the Ankeny-Artin-Chowla (AAC) conjecture is reported. As a result, in the case of primes p which are ≡ 5 mod 8, the AAC conjecture has been verified using a different multiple of the regulator of the quadratic field ℚ(Formula Presented) than was meant. However, since any multiple of this regulator is suitable for this purpose, provided that it is smaller than 8p, the main result that the AAC conjecture is true for all the primes ≡ 1 mod 4 which are 1011, remains valid. As an addition, we have verified the AAC conjecture for all the primes ≡ 1 mod 4 between 1011 and 2 × 1011, with the corrected program