English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Let      d 1 , d 2 {d_1},\;{d_2}       be coprime positive integers, which are squarefree, and let      h h       denote the class number of the imaginary quadratic field      Q ( −  d 1 d 2 ) \mathbb {Q}(\sqrt { - {d_1}{d_2}} )      . Let      m , n m,\;n       be integers such that      m ⩾  0 , n &gt; 1 m \geqslant 0,\;n &gt; 1      , and      gcd ( n , 2 h ) = 1 \gcd (n,2h) = 1      . In this paper we prove that if      n ⩾  8.5 ⋅  10 6 n \geqslant 8.5 \cdot {10^6}      , then the equations      d 1 x 2 + 2 2 m d 2 = y n ( 2 ∤  y ) {d_1}{x^2} + {2^{2m}}{d_2} = {y^n}(2\nmid y)       and      d 1 x 2 + d 2 = 4 y n {d_1}{x^2} + {d_2} = 4{y^n}       have no positive integer solutions      ( x , y ) (x,y)       with      gcd ( x , y ) = 1 \gcd (x,y) = 1      .

On the Diophantine equations 𝑑₁𝑥²+2^{2𝑚}𝑑₂=𝑦ⁿ and 𝑑₁𝑥²+𝑑₂=4𝑦ⁿ