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512 is the smallest (and only known) cube such that 512*2^5121 is prime. [Gupta]
512 = 2^9. Both 2^9 +/ 9^2 are prime. This is the smallest number of the form 2^n such that both 2^n +/ n^2 are prime. [Astle]
(There is one curio for this number that has not yet been approved by an editor.)
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