Let n = pq, p, q distinct odd primes. We can divide the numbers in Zn into four classes depending on their membership in QRp and QRq. Let Q11 n be those numbers that are quadratic residues mod both p and q; let Q10 n be those numbers that are quadratic residues mod p but not mod q; let Q 01 n be those numbers that are quadratic residues mod q but not mod p; and let Q00 n be those numbers that a...

Fermat’s little theorem says for prime p that ap−1 ≡ 1 mod p for all a 6≡ 0 mod p. A naive extension of this to a composite modulus n ≥ 2 would be: for a 6≡ 0 mod n, an−1 ≡ 1 mod n. Let’s call this “Fermat’s little congruence.” It may or may not be true. When n is prime, it is true for all a 6≡ 0 mod n. But when n is composite it usually has many counterexamples. Example 1.1. When n = 15, the t...

On primitive roots of 1 mod p k , divisors of p 2 − 1 , Wieferich primes and quadratic analysis mod p Abstract Primitive roots of 1 mod p k (k > 2 and odd prime p) are sought, in cyclic units group G k ≡ A k B k mod p k , coprime to p, of order (p − 1)p k−1. 'Core' subgroup A k has order p − 1 independent of precision k, and 'extension' subgroup B k of all p k−1 residues 1 mod p is generated by...

On primitive roots of 1 mod p k , divisors of p ± 1, Wieferich primes, and quadratic analysis mod p Abstract Primitive roots of 1 mod p k (k >2 and odd prime p) are sought, in cyclic units group G k ≡ A k B k mod p k , coprime to p, of order (p − 1)p k−1. 'Core' subgroup A k has order p − 1 independent of precision k, and 'extension' subgroup B k of all p k−1 residues 1 mod p is generated by p+...

The structural and thermodynamic features of three polymorphic forms of the local anesthetic drug oxybuprocaine hydrochloride (OBPHC) were characterized by hot-stage microscopy, differential scanning calorimetry (DSC), pycnometry, Fourier transform infrared spectroscopy (FTIR), FT-Raman and solid-state NMR (SSNMR) spectroscopy as well as X-ray powder and single-crystal diffractometry. Mod II° c...

Recall the collection of functions {BlumN : QRN → QRN} were BlumN = x mod N for so-called Blum integers N that are products of two distinct primes congruent to 3 mod 4 are candidate one-way functions. As we mentioned earlier, inverting for algorithms for BlumN yield algorithms for factoring N . We now show that for N = pq, p 6= q prime, p, q ≡ 3 (mod 4), the pair (p, q) of factors of N forms tr...

Modafinil (MOD) exhibits therapeutic efficacy for treating sleep and psychiatric disorders; however, its mechanism is not completely understood. Compared with other psychostimulants inhibiting dopamine (DA) uptake, MOD weakly interacts with the dopamine transporter (DAT) and modestly elevates striatal dialysate DA, suggesting additional targets besides DAT. However, the ability of MOD to induce...

A 3 copies of the manuscript A please indicate the corresponding author's details on the manuscript (tel/fax/e-mail) A 3 sets of figures A electronic version of the manuscript (Label the floppy disk with the type of computer and word processing package used, your name and the name of the file on the disk.) A electronic version of the figures For Review Papers: A 3 copies of the manuscript A ple...

A 3 copies of the manuscript A please indicate the corresponding author's details on the manuscript (tel/fax/e-mail) A 3 sets of figures A electronic version of the manuscript (Label the floppy disk with the type of computer and word processing package used, your name and the name of the file on the disk.) A electronic version of the figures For Review Papers: A 3 copies of the manuscript A ple...