0.1.3 Unique Factorization Theorem If a is an integer, not 0 or ±1, then (1) a can be written as a product p1 · · · pn of primes. (2) If a = p1 · · · pn = q1 · · · qm, where the pi and qj are prime, then n = m and, after renumbering, pi = ±qi for all i. [We allow negative primes, so that, for example, −17 is prime. This is consistent with the general definition of prime element in an integral d...

Let X 0 (N) be the Atkin-Lehner quotient of the modular curve X0(N) associated to the Fricke involution wN . Assume N ≥ 5 prime and endow the real locus X 0 (N)(R) with the real topology. In this paper we revisit a special case of a result due to Ogg on the connected components of X 0 (N)(R). Then we obtain a formula for the homology class of each connected component of X 0 (N)(R) in terms of M...

In this paper, we mainly prove the following conjecture of Z.-H. Sun cite{SH20}: Let $p>3$ be a prime. Then $$\sum_{k=0}^{p-1}\binom{2k}k\frac{3k+1}{(-16)^k}f_k\equiv(-1)^{(p-1)/2}p+p^3E_{p-3}\pmod{p^4},$$ where $f_n=\sum_{k=0}^n\binom{n}k^3$ and $E_n$ stand for $n$th Franel number Euler respectively.

Using elementary methods, the following results are obtalned:(1) If p is n n-m prime, 0 m n, 0 < b < ap n-m, and p ab, then (m) (-I)P-I(apb (rood pn). 2 A2_4B If r,s are the roots of x Ax-B, where (A,B) and D > 0, if n n u v rn+s n, and k > O, then (II) v =v (rood pn). n r-s n kpn kpn-1 (III) If p is odd and p D, then u (_D) u (rood pn); kp n P kp n-I (IV) u (_1)Bu n) n n-I (rood 2 k2 k2

In this paper, we study the Diophantine equation $(p+4n)^x+p^y=z^2,$ where $n$ is a non-negative integer and $p, p+4n$ are prime numbers such that $p\equiv 7\pmod{12}$. We show solutions of $(x, y, z)\in \{(0, 1, \sqrt {p+1})\} \cup \{ (1, 0, 2\sqrt{n+\frac{p+1}{4}})\}$, $\sqrt {p+1}$ $\sqrt{n+\frac{p+1}{4}}$ integers.

We show that for any fixed ε > 0, there are numbers δ > 0 and p 0 2 with the following property: for every prime p p 0 , there is an integer p δ < N p 1/(4 √ e)+ε such that the sequence 1, 2,. .. , N contains at least δN quadratic non-residues modulo p. We then apply this result to obtain a new estimate on the smallest quadratic nonresidue in a Beatty sequence.

Here we introduce a simple elementary method due to myself. Let n be nk(p) or nk(p) − 1 according to whether −1 is a kth power residue mod p. Note that −n is a kth power nonresidue mod p. For i = 1, · · · , nk(p) − 1, clearly p − in is a kth power nonresidue mod p; if p − in > 0 then we must have p− in > nk(p) > n and hence p− (i+ 1)n > 0 since p is a prime. As p− n > 0, by the above p− nk(p)n ...

‎Let $p$ be a prime with $pgeq 7$ and $q=2(p-1)$‎. ‎In this paper‎ ‎we prove the existence of a nontrivial product of‎ ‎filtration $s+4$ in the stable homotopy groups of spheres‎. ‎This nontrivial‎ ‎product is shown to be represented up to a nonzero scalar by‎ ‎the product element $widetilde{gamma}_{s}b_{n-1}g_{0}in‎ ‎{Ext}_{mathcal{A}}^{s+4,(p^n+sp^2+sp+s)q+s-3}(mathbb{Z}/p,mathbb{Z}/p)$‎ ‎in ...