On equations $(-1)^αp^x+(-1)^β(2^k(2p+1))^y=z^2$ with Sophie Germain prime $p$
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2022
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| author | Li, Yuan Zhang, Jing Liu, Baoxing |
| author_facet | Li, Yuan Zhang, Jing Liu, Baoxing |
| contents | 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 $k\in\{1,3,5,7\}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2212_01670 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On equations $(-1)^αp^x+(-1)^β(2^k(2p+1))^y=z^2$ with Sophie Germain prime $p$ Li, Yuan Zhang, Jing Liu, Baoxing Number Theory 11A15, 11D61, 11D72, 14H52 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 $k\in\{1,3,5,7\}$. |
| title | On equations $(-1)^αp^x+(-1)^β(2^k(2p+1))^y=z^2$ with Sophie Germain prime $p$ |
| topic | Number Theory 11A15, 11D61, 11D72, 14H52 |
| url | https://arxiv.org/abs/2212.01670 |