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We analyze the Brauer–Manin obstruction to rational points on the K3 surfaces over $${{\mathbb {Q}}}$$ Q given by double covers of $${{\mathbb {P}}^{2}}$$ P2 ramified over a diagonal sextic. After finding an explicit set of generators for the geometric Picard group of such a surface, we find two types of infinite families of counterexamples to the Hasse principle explained by the algebraic Brauer–Manin...
Inspired by the Numberphile video “The uncracked problem with 33” by Browning and Brady Haran (https://youtu.be/wymmCdLdPvM), we investigate solutions to $$x^3+y^3+z^3=k$$ x3+y3+z3=k for a few small values of k. We find the first known solutions for $$k=33$$ k=33 and $$k=795$$ k=795 .
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