Are there infinitely many $N^3$ (especially for prime $N$) that cannot be expressed as a sum of three positive cubes?"

number-theory

Don't know about prime $N,$ but otherwise yes: http://mathworld.wolfram.com/DiophantineEquation3rdPowers.html

Related

Analytic form of: $ \int \frac{\bigl[\cos^{-1}(x)\sqrt{1-x^2}\bigr]^{-1}}{\ln\bigl( 1+\sin(2x\sqrt{1-x^2})/\pi\bigr)} dx $
Painting the plane red and blue: Is it possible for each unit circumference to contain exactly $n$ blue points?
equidistant points in $\mathbb{R}^n$ equipped with $\|.\|_p$
How can there be alternatives for the foundations of mathematics?
Inverse of a Toeplitz matrix
Seeking Fourier series solution on Laplace equation...still looking, am I on track?
If $\beta=0.{a_1}^{k}{a_2}^{k}{a_3}^{k}\cdots\in\mathbb Q$, then $\alpha=0.a_1a_2a_3\cdots\in\mathbb Q$?
Eigenvalues and eigenvectors of Hadamard product of two positive definite matrices
Explicit description of small open set containing the rationals
Is there an infinite-dimensional Jordan decomposition?
Mathematics Audiobooks for the Blind?
Function defined as a limit