On Computations of Shanks and Schmid

PRINCIPAL FORMS X 2 + nY 2 REPRESENTING MANY INTEGERS

In 1966, Shanks and Schmid investigated the asymptotic behavior of the number of positive integers less than or equal to x which are represented by the quadratic form X +nY . Based on some numerical computations, they observed that the constant occurring in the main term appears to be the largest for n = 2. In this paper, we prove that in fact this constant is unbounded as n runs through positi...

Variations on a Theorem of Landau . Part I By Daniel Shanks and

Congenital Adrenal Hyperplasia and Schmid Metaphyseal Chondrodysplasia in a Child

Congenital adrenal hyperplasia (CAH) is a group of hereditary diseases, which are autosomal recessive. CAH occurs due to defect in one of the cortisol coding genes and often clinically presents itself with signs of androgen overproduction. In this article, we report a case of CAH and Schmid metaphyseal dysplasia. Our literature review indicated that this report is the first attempt on CYP11B1 a...

A High-Speed Square Root Algorithm in Extension Fields

A square root (SQRT) algorithm inGF (p) (m=r0r1 · · · rn−12, ri: odd prime, d>0: integer) is proposed in this paper. First, the TonelliShanks algorithm is modified to compute the inverse SQRT in GF (p d ), where most of the computations are performed in the corresponding subfields GF (p i ) for 0 i d − 1. Then the Frobenius mappings with an addition chain are adopted for the proposed SQRT algor...

A High - Speed Square Root Algorithm for Extension fields – Especially for Fast Extension Fields –

A square root (SQRT) algorithm in extension field Fpm(m = r0r1 · · · rn−1 · 2, ri : odd prime, d : positive integer) is proposed in this paper. First, a conventional SQRT algorithm, the TonelliShanks algorithm, is modified to compute the inverse SQRT in F p2 , where most of the computations are performed in the corresponding subfields Fp2i for 0 6 i 6 d − 1. Then the Frobenius mappings with add...