Let N(a, m) be the least integer n (if exists) such that φ(n) ≡ a (mod m). Friedlander and Shparlinski proved that for any ε > 0 there exists A = A(ε) > 0 such that for any positive integer m which has no prime divisors p < (log m) and any integer a with gcd(a,m) = 1, we have the bound N(a,m) ¿ m. In the present paper we improve this bound to N(a,m) ¿ m.

Let q = p with p be an odd prime. Let 0 ≤ k ≤ n − 1 and k 6= n/2. In this paper we determine the value distribution of following exponential(character) sums

Schrag and Crawford ( 1996) present strong experimental evidence that the occurrence of prime implicates of varying lengths in random instances of 3-SAT exhibits behaviour similar to the wellknown phase transition phenomenon associated with satisfiability. Thus, as the ratio of number of clauses (m) to number of propositional variables (n) increases, random instances of 3-SAT progress from form...

‎‎In this paper‎, ‎we define fuzzy subnexuses over a nexus $N$‎. ‎Define and study the notions of the prime fuzzy subnexuses and the fractions‎‎induced by them‎. ‎Finally‎, ‎we show that if S is a meet‎‎closed subset of the set Fsub(N), ‎of fuzzy subnexuses of a nexus N‎, ‎and‎‎h= ⋀S ϵ S, ‎then the fractions S^-1 N and h^-1 N are isomorphic as meet-semilattices‎.

Let $p$ be an odd prime and let $n$ a positive integer with $p\nmid n$. For any $r$ $\lambda \in \{1, 2, 3\}$ \lambda $, we have $$ \sum _{k=0}^{p^{r}n-1}\frac {\left (\frac 12\right )_k}{k!}\cdot \frac {4^k}{\lambda ^

We raise the following new conjecture: Each natural number n 6= 216 can be written in the form p + Tx, where p is 0 or a prime, and Tx = x(x + 1)/2 is a triangular number. Some extensions and variants of this conjecture are also discussed. 1. On integers of the form p + Tx Since 1+ 2+ · · ·+ n = n(n+1)/2, those integers Tx = x(x+1)/2 with x ∈ Z are called triangular numbers. Here is a list of t...

For various positive integers k, the sums of kth powers of the first n positive integers, Sk(n) := 1 k+2k+ · · ·+nk, are some of the most popular sums in all of mathematics. In this note we prove a congruence modulo n3 involving two consecutive sums S2k(n) and S2k+1(n). This congruence allows us to establish an equivalent formulation of Giuga’s conjecture. Moreover, if k is even and n ≥ 5 is a ...

We observe some new characterizations of $n$-presented modules. Using the concepts of $(n,0)$-injectivity and $(n,0)$-flatness of modules, we also present some characterizations of right $n$-coherent rings, right $n$-hereditary rings, and right $n$-regular rings.