For every positive integer n let p(n) be the largest prime number p ≤ n. Given a positive integer n = n1, we study the positive integer r = R(n) such that if we define recursively ni+1 = ni − p(ni) for i ≥ 1, then nr is a prime or 1. We obtain upper bounds for R(n) as well as an estimate for the set of n whose R(n) takes on a fixed value k.

For n = 1, 2, 3, . . . define S(n) as the smallest integer m > 1 such that those 2k(k − 1) mod m for k = 1, . . . , n are pairwise distinct; we show that S(n) is the least prime greater than 2n− 2 and hence the value set of the function S(n) is exactly the set of all prime numbers. For every n = 4, 5, . . . , we prove that the least prime p > 3n with p ≡ 1 (mod 3) is just the least positive int...

In this paper, we find all integers x such that x − 1 has only prime factors smaller than 100. This gives some interesting numerical corollaries. For example, for any positive integer n we can find the largest positive integer x such that all prime factors of each of x, x + 1, . . . , x+ n are less than 100.

We study whether sufficiently large integers can be written in the form cp+ Tx, where p is either zero or a prime congruent to r mod d, and Tx = x(x + 1)/2 is a triangular number. We also investigate whether there are infinitely many positive integers not of the form (2ap−r)/m+Tx with p a prime and x an integer. Besides two theorems, the paper also contains several conjectures together with rel...

We study whether sufficiently large integers can be written in the form cp+ Tx, where p is either zero or a prime congruent to r mod d, and Tx = x(x + 1)/2 is a triangular number. We also investigate whether there are infinitely many positive integers not of the form (2p−r)/m+Tx with p a prime and x an integer. Besides two theorems, the paper also contains several conjectures together with rela...

vasil'ev posed problem 16.26 in [the kourovka notebook: unsolved problems in group theory, 16th ed.,sobolev inst. math., novosibirsk (2006).] as follows:does there exist a positive integer $k$ such that there are no $k$ pairwise nonisomorphicnonabelian finite simple groups with the same graphs of primes? conjecture: $k = 5$.in [zvezdina, on nonabelian simple groups having the same prime gr...

Let p be a prime number, and s a positive integer. Let k be an integer, and let n be an integer such that n ≥ k ≥ p s − p s−1 .

let $g={rm sl}_2(p^f)$ be a special linear group and $p$ be a sylow‎ ‎$2$-subgroup of $g$‎, ‎where $p$ is a prime and $f$ is a positive‎ ‎integer such that $p^f>3$‎. ‎by $n_g(p)$ we denote the normalizer of‎ ‎$p$ in $g$‎. ‎in this paper‎, ‎we show that $n_g(p)$ is nilpotent (or‎ ‎$2$-nilpotent‎, ‎or supersolvable) if and only if $p^{2f}equiv‎ ‎1,({rm mod},16)$‎.

let $p$ be a prime number and $n$ be a positive integer. the graph $g_p(n)$ is a graph with vertex set $[n]={1,2,ldots ,n}$, in which there is an arc from $u$ to $v$ if and only if $uneq v$ and $pnmid u+v$. in this paper it is shown that $g_p(n)$ is a perfect graph. in addition, an explicit formula for the chromatic number of such graph is given.