In this paper we confirm a conjecture of Sun which states that each positive integer is a sum of a square, an odd square and a triangular number. Given any positive integer m, we show that p = 2m + 1 is a prime congruent to 3 modulo 4 if and only if Tm = m(m + 1)/2 cannot be expressed as a sum of two odd squares and a triangular number, i.e., p = x+8(y+z) for no odd integers x, y, z. We also sh...

Definition 2.1. Suppose p1, . . . , pk are polynomials in F[x] which are not all 0. Set I = 〈p1, . . . , pk〉. Let d denote the monic generator of I. We call d the greatest common divisor of the pi and write d = gcd(p1, . . . , pk). If d = 1, we say that the polynomials p1, . . . , pk is relatively prime. Note that, by definition, there exists polynomials q1, . . . , qk ∈ F[x] such that d = p1q1...

A k×n Latin rectangle on the symbols {1, 2, . . . , n} is called reduced if the first row is (1, 2, . . . , n) and the first column is (1, 2, . . . , k) . Let Rk,n be the number of reduced k × n Latin rectangles and m = bn/2c. We prove several results giving divisors of Rk,n. For example, (k − 1)! divides Rk,n when k ≤ m and m! divides Rk,n when m < k ≤ n. We establish a recurrence which determ...

G. Giuga conjectured that if an integer n satisses n?1 P k=1 k n?1 ?1 mod n, then n must be a prime. We survey what is known about this interesting and now fairly old conjecture. Giuga proved that n is a counterexample to his conjecture if and only if each prime divisor p of n satisses (p ? 1) j (n=p ? 1) and p j (n=p ? 1). Using this characterization, he proved computationally that any counter...

Let K = k(B) be a function field of an integral projective variety B over the algebraically closed field k such that B is regular in codimension 1. The set of places MB is given by the prime divisors of B. We fix an ample class c on B. If we count every prime divisor Y with weight deg c (Y ), then the valuations ordY lead to a product formula on K and hence to a theory of heights (see [La] or [...

Given a well-chosen additively homomorphic cryptosystem and a Σ protocol with a linear answer, Damg̊ard, Fazio, and Nicolosi proposed a non-interactive designated-verifier zero knowledge argument in the registered public key model that is sound under non-standard complexity-leveraging assumptions. In 2015, Chaidos and Groth showed how to achieve the weaker yet reasonable culpable soundness notio...

Let H be a Krull monoid with finite class group G such that every contains prime divisor. We consider the system L(H) of all sets lengths and study when or is contained in L(H?) H? G?, divisors classes Davenport constant D(G?)=D(G). Among others, we show if either cyclic order m?7 an elementary 2-group rank m?1?6, G? any which non-isomorphic to but D(G?)=D(G), then systems are incomparable.

dpH (G,EL) = dpH (G,Z/pZ)− dpH (G,Z/pZ) + dpH (G,Z/pZ). It is so crucial to find an upperbound for the p-rank dpH (G,EL) when Cl(L) is trivial. In this paper, we prove results about this rank in some special cases. More precisely, we compute this p-rank when L/K is an abelian unramified (also at infinity) p-extension whose Galois group can be generated by two elements. We also exhibit an explic...

Problem 3-1. Prime Triangles. TA Notes: It is a little-known fact (to us, anyway) that the Java 1.4.1 class library contains a bug in part of its primality tester. Specifically, the Lucas-Lehmer test is implemented incorrectly, and sometimes reports that certain large numbers are composite when they are actually prime. This slowed down our own trianglefinding algorithm for large values of k, an...

We study the deformation theory of rational curves on primitive symplectic varieties and show that if cover a divisor, then, as in smooth case, they deform along their Hodge locus universal locally trivial deformation. As applications, we extend Markman's invariance prime exceptional divisors to this singular framework provide existence results for uniruled ample which are deformations any modu...