论文标题
在等式上$（-1）^αp^x+（-1）^β（2^k（2p+1））
On equations $(-1)^αp^x+(-1)^β(2^k(2p+1))^y=z^2$ with Sophie Germain prime $p$
论文作者
论文摘要
在本文中，我们考虑了diophantine方程$（-1）^αp^x+（-1）^β（2^k（2p+1））^y = z^2 $ for Sophie Germain Prime $ p $，$α，β\ in \ in \ in \ {0,1 \}，$ {0,1 \}，$αβ= 0 $ $αβ= 0 $和$ K \ geq \ geq 0 $。首先，对于$ p = 2 $，我们通过使用nagell-lijunggren方程和椭圆形曲线的数据库lmfdb $ y^2 = y^2 = x^3+b $ by $ p = y^2 = x^3+b $ by $ p $ y nagell-lijunggren方程$（-1）^α2^α2^x+（-1）^β（2^{k} 5）^y = Z^2 $ $然后，我们为以下四种类型的方程式获得了所有非负整数解决方案，用于奇怪的Sophie Germain Prime $ p $： i）$ p^x+（2^{2k+1}（2p+1））^y = z^2 $，带有$ p \ equiv 3，5 \ pmod 8 $和$ k \ geq 0 $; ii）$ p^x+（2^{2k}（2p+1））^y = z^2 $ with $ p \ equiv 3 \ pmod 8 $和$ k \ geq 1 $; iii）$ p^x-（2^{k}（2p+1））^y = z^2 $ with $ p \ equiv 3 \ pmod 4 $和$ k \ geq 0 $; iv）$ -p^x+（2^{k}（2p+1））^y = z^2 $，带有$ p \ equiv 1，3，3，5 \ pmod 8 $和$ k \ geq 1 $; 对于每种类型的方程式，我们显示了这种Prime $ P $的存在。由于猜想文学中存在无限的索菲·埃格曼（Sophie Germain）素数，因此可以合理地猜想存在无限的索菲·埃格曼（Sophie Germain）素质$ p $ p $ p $ p $ p $ p \ equiv k \ pmod 8 $，用于任何$ k \ in \ in \ in \ {1,3,5,7 \} $。
In this paper, we consider the Diophantine equation $(-1)^αp^x+(-1)^β(2^k(2p+1))^y=z^2$ for Sophie Germain prime $p$ with $α, β\in\{0,1\}$, $αβ=0$ and $k\geq 0$. First, for $p=2$, we solve three Diophantine equations $(-1)^α2^x+(-1)^β(2^{k} 5)^y=z^2$ by using Nagell-Lijunggren Equation and the database LMFDB of elliptic curve $y^2=x^3+ax+b$ over $\mathbb{Q}$. Then we obtain all non-negative integer solutions for the following four types of equations for odd Sophie Germain prime $p$: i) $p^x+(2^{2k+1}(2p+1))^y=z^2$ with $p\equiv 3, 5 \pmod 8$ and $k\geq 0$; ii) $p^x+(2^{2k}(2p+1))^y=z^2$ with $p\equiv 3 \pmod 8$ and $k\geq 1$; iii) $p^x-(2^{k}(2p+1))^y=z^2$ with $p\equiv 3 \pmod 4$ and $k\geq 0$; iv) $-p^x+(2^{k}(2p+1))^y=z^2$ with $p\equiv 1, 3, 5 \pmod 8$ and $k\geq 1$; For each type of the equations, we show the existences of such prime $p$. Since it was conjectured that there exist infinitely many Sophie Germain primes in literature, it is reasonable to conjecture that there exist infinite Sophie Germain primes $p$ such that $p\equiv k \pmod 8$ for any $k\in\{1,3,5,7\}$.
