Intereting Posts

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Smash product of compact spaces
Prove $\int_{0}^{\infty}{1\over x}\cdot{1-e^{-\phi{x}}\over 1+e^{\phi{x}}}dx=\ln\left({\pi\over 2}\right)$
What is a rigorous proof of the topological equivalence between a donut and a coffee mug?
How many 2-edge-colourings of $K_n$ are there?
Is $\mathbb{Q}(\sqrt{2}) \cong \mathbb{Q}(\sqrt{3})$?
Area form for $M^2 \subseteq \Bbb R^4$
GRE – Probability Question
Difference between Gâteaux and Frêchet derivatives
Convergence of Lebesgue integrals
Neighborhoods: Interior
A certain “harmonic” sum
What is the definition of a complex manifold with boundary?
Can a cubic equation have three complex roots?
another balls and bins question

Primes of the form $p=4k+1\;$ have a unique decomposition as sum of squares $p=a^2+b^2$ with $0<a<b\;$, due to Thue’s Lemma.

What is known about sums of $n$ higher powers resulting in primes?

I tried $a^3+b^3+c^3$, asked Wolfram and found $3,17,29,43,73$, a sequence that I don’t know (EDIT: thanks to Matthew: A007490). Interestingly, that $251$ has even $2$ decompositions, $1^3+5^3+5^3=2^3+3^3+6^3$.

- The square of n+1-th prime is less than the product of the first n primes.
- Contradiction: Prove 2+2 = 5
- Determine the divisibility of a given number without performing full division
- A game with two dice
- How come if $\ i\ $ not of the following form, then $12i + 5$ must be prime?
- Prove: The positive integers cannot be partitioned into arithmetic sequences (using Complex Analysis)

Can anybody help here?

- Is the algebraic closure of a $p$-adic field complete
- Finding primes so that $x^p+y^p=z^p$ is unsolvable in the p-adic units
- Dirichlet Series and Average Values of Certain Arithmetic Functions
- Properties of squares in $\mathbb Q_p$
- Bunyakovsky conjecture for cyclotomic polynomials
- Gaps between numbers of the form $pq$
- Solving a Word Problem relating to factorisation
- A condition for being a prime: $\;\forall m,n\in\mathbb Z^+\!:\,p=m+n\implies \gcd(m,n)=1$
- Simple explanation and examples of the Miller-Rabin Primality Test
- Proving the converse of an IMO problem: are there infinitely many pairs of positive integers (m,n) such that m divides n^2 + 1 and n divides m^2 + 1?

Any cube is congruent to $0$, $1$, or $-1$ modulo $9$. It follows that the sum of three cubes cannot be congruent to $4$ or $5$ modulo $9$. So we have a congruential restriction analogous to the case of two squares.

By Dirichlet’s Theorem on primes in arithmetic progressions (or undoubtedly by much more elementary means) one can show that there are infinitely many primes congruent to $4$ modulo $9$, and also infinitely many primes congruent to $5$. Thus there are infinitely many primes that cannot be represented as the sum of three cubes.

About primes not congruent to $4$ or $5$ modulo $9$, one cannot say much. A long-standing conjecture is that *every* integer which is not congruent to $4$ or $5$ modulo $9$ is the sum of $3$ cubes. (Here negative cubes are allowed.) There has been a lot of computation on this problem.

For other powers, one should go to the vast literature on Waring’s Problem.

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- Finding center and radius of circumcircle