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I'm looking for a result (embarrassingly enough, a somewhat famous result) which shows the infinitude in some sense I don't recall of primes of the form $$ \lfloor x^k\rfloor $$ for $k$ fixed and irrational. There were sharp limits on the size of $k.$ I think the original result has been improved many times, mostly by widening the allowable range of $k.$

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It's somewhat similar to Alkauskas & Dubickas 2004 (though not that new), in case that helps give the flavor of the result. –  Charles Jun 13 '12 at 18:51
    
Presumably you want $x=2,3,4,5,\dots$? –  Gerry Myerson Jun 14 '12 at 0:25
    
The Alkauskas-Dubickas result has varying integer exponents, you're after fixed irrational exponents, so I'd say it's not that similar. –  Gerry Myerson Jun 14 '12 at 0:27
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1 Answer

up vote 4 down vote accepted

Rivat and Sargos, Nombres premiers de la forme $[n^c]$, Canad. J. Math. 53 (2001), no. 2, 414–433, MR1820915 (2002a:11107), reviewed by G. Greaves.

The authors establish an asymptotic formula for the number of primes not exceeding $x$ of the form $[n^c]$. Their result applies for each $c$ with $1\lt c\lt2817/2426$. The review compares this to previous work, and there are links to other papers and reviews that cite this paper.

Apparently the first paper along these lines was by Piatetski-Shapiro in 1953, with $1\lt c\lt12/11$.

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That's it. I've heard it called the "Piatetski-Shapiro Prime Number Theorem" because of its density result: $x/c\log x.$ –  Charles Jun 14 '12 at 13:25
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