A 5-cycle system on v + w points embeds a balanced P4-design on v points if there is a subset of v points on which the 5-cycles induce the blocks of a balanced P4-design. The mininum possible such w is: v (mod 30) w v ≡ 4, 16 v−1 3 v ≡ 7, 10, 22, 25 v+5 3 v ≡ 1, 13, 28 v+11 3 v ≡ 19 (mod 30) v+17 3 We shall show this minimum value of w is attained. AMS classification: 05B05.

Cyclic m-cycle systems of order v are constructed for all m ≥ 3, and all v ≡ 1(mod 2m). This result has been settled previously by several authors. In this paper, we provide a different solution, as a consequence of a more general result, which handles all cases using similar methods and which also allows us to prove necessary and sufficient conditions for the existence of a cyclic m-cycle syst...

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...

We establish congruences for higher order Euler polynomials modulo 720. apply this result constructing analogues of Stern secant numbers $E_{4n}\equiv 5(\mathrm{mod}\ 60), E_{4n+2}\equiv -1(\mathrm{mod}\ 60)$ to tangent and Genocchi numbers. prove that satisfy the following $E_{4n+1}\equiv 16(\mathrm{mod}\ 720)$, $E_{4n+3}\equiv -272(\mathrm{mod}\ 720)$. 12-periodic property 45.

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...

With the exploration of the MA plasma current regime in up to 0.5 s long discharges, RFX-mod has opened new and very promising perspectives for the Reversed Field Pinch (RFP) magnetic configuration, and has made a significant progress in understanding and improving confinement and in controlling plasma stability. A big leap with respect to previous knowledge and expectations on RFP physics and ...

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...