. Let A ⊂ R > 0 be a ﬁnite set of distances, and let G A ( R n ) be the graph with vertex set R n and edge set { ( x, y ) ∈ R n : k x − y k 2 ∈ A } , and let χ ( R n , A ) = χ ( G A ( R n )) . Erdős… Expand

Let p n be the n -th prime, and deﬁne the maximal prime gap G ( x ) as G ( x ) = max p n ≤ x ( p n +1 − p n ) . We give a summary of the lower and upper bounds that have been obtained for G ( x )… Expand

The following is proven using arguments that do not revolve around the Riemann Hypothesis or Sieve Theory. If $p_n$ is the $n^{\rm th}$ prime and $g_n=p_{n+1}-p_n$, then $g_n=O({p_n}^{2/3})$.

Consider two consecutive odd primes $p_n$ and $p_{n+1}$, let $m$ to be their midpoint, fixed once for all. We prove unconditionally that every $x$ in the interval $[\frac{\ln{(m-p_n)}}{\log{p_n}},… Expand

The Legendre conjecture has resisted analysis over a century, even under assumption of the Riemann Hypothesis. We present, a significant improvement on previous results by greatly reducing the… Expand

Let $n\in\mathbb{Z}^+$. Is it true that every sequence of $n$ consecutive integers greater than $n^2$ and smaller than $(n+1)^2$ contains at least one prime number? In this paper we show that this is… Expand

Update: This work reproduces an earlier result of Peck, which the author was initially unaware of. The method of the proof is essentially the same as the original work of Peck. There are no new… Expand

It is shown that the problem of determining rep(K 1,n ) is equivalent to determining the smallest even k for which φ(k) ≥ n: this problem for “small” n is solved and the possible forms of rep( K 1, n ) for sufficiently large n are determined.Expand

A graph G has a representation modulo r if there exists an injective map f : V (G) → {0, 1, . . . , r−1} such that vertices u and v are adjacent if and only if f(u)−f(v) is relatively prime to r. The… Expand