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Let V be a valuation domain and let A=V+εV be a dual valuation domain. We propose a method for computing a strong Gröbner basis in R=A[x1,…,xn]; given polynomials f1,…,fs∈R, a method for computing a generating set for Syz(f1,…,fs)={(h1,…,hs)∈Rs∣h1f1+⋯+hsfs=0} is given; and, finally, given two ideals I=〈f1,…,fs〉 and J=〈g1,…,gr〉 of R, we propose an algorithm for computing a generating set for I∩J.

In this paper, we study algebraic properties of lattice points of the arc on the conics x 2 − d y 2 = N especially for d = 1 , which is the Fermat factorization equation that is the main idea of many important factorization methods like the quadratic field sieve, using arithmetical results of a particular hyperbola parametrization. As a result, we present a generalization of the forms, the cardinal, and the distribution of its lattice points over the integers. In particular, we prove that if N − 6 ≡ 0   mod   4 , Fermat’s method fails. Otherwise, in terms of cardinality, it has, respectively, 4, 8, 2 α + 1 , 1 − δ 2 p i 2 n + 1 , and 2 ∏ i = 1 n α i + 1 lattice pointts if N is an odd prime, N = N a × N b with N a and N b being odd primes, N = N a α with N a being prime, N = ∏ i = 1 n p i with p i being distinct primes, and N = ∏ i = 1 n N i α i with N i being odd primes. These results are important since they provide further arithmetical understanding and information on the integer solutions revealing factors of N . These results could be particularly investigated for the purpose of improving the underlying integer factorization methods.

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