Intereting Posts

Show that $(f_n)$ is equicontinuous, given uniform convergence
How to prove $D^n/S^{n-1}\cong S^n$?
The number of ways of writing an integer as a sum of two squares
Are all manifolds in the usual sense also “vector manifolds”?
Question on groups of order $pq$
Find solution of equation $(z+1)^5=z^5$
Derived sets – prove $(A \cup B)' = A' \cup B'$
Prove that $\dfrac{\pi}{\phi^2}<\dfrac{6}5 $
How to prove that continuous functions are Riemann-integrable?
All natural solutions of $2x^2-1=y^{15}$
What comes after tetration ? And after ? And after ? etc.
Moore space, induced map in homology
Questions about definition of Tangent Space
Does an injective endomorphism of a finitely-generated free R-module have nonzero determinant?
For $n \in \mathbb{N}$ $\lfloor{\sqrt{n} + \sqrt{n+1}\rfloor} = \lfloor{\sqrt{4n+2}\rfloor}$

I’m refreshing some algebra and in the beginning chapters of my book I’m presented with the following simple looking problem that I got stuck on:

Prove or disprove: If $n$ is a positive integer, then $n=p+a^2$, where $a$ is an integer and $p$ is prime or $p=1$.

At first I thought $n=13$ would give a nice counterexample, but then I noticed my book defines primes to include the negative primes. Now $13=-3+4^2$ is a solution.

- Show that $8 \mid (a^2-b^2)$ for $a$ and $b$ both odd
- A function can provide the complete set of Euler primes via a Mill's-like constant. Is it useful or just a curiosity?
- Is this Dirichlet series generating function of the von Mangoldt function matrix correct?
- Number of Multiples of $10^{44}$ that divide $ 10^{55}$
- $2^{4n+1} \equiv 1 \pmod{8n+7}$, this has been bugging me
- Help understand the proof of infinitely many primes of the form $4n+3$

I am allowed to use the fundamental theorem of arithmetic and the division and euclidean algorithms. I tried to express $n$ and $a$ by their prime factorizations, but I didn’t get anywhere from there. I would like a hint which direction to take. Am I just missing something silly?

- how to calculate double sum of GCD(i,j)?
- GCD Proof with Multiplication: gcd(ax,bx) = x$\cdot$gcd(a,b)
- Concise proof that every common divisor divides GCD without Bezout's identity?
- Connections between number theory and abstract algebra.
- Primes congruent to 1 mod 6
- If $d>1$ is a squarefree integer, show that $x^2 - dy^2 = c$ gives some bounds in terms of a fundamental solution.
- How to show $\binom{2p}{p} \equiv 2\pmod p$?
- Is there a conjecture with maximal prime gaps
- Accuracy of Fermat's Little Theorem?
- Is this Dirichlet series generating function of the von Mangoldt function matrix correct?

Try $n=169$. The number $169-a^2$ factors as $(13-a)(13+a)$. Can one of these factors be $\pm 1$ and the other “prime”? The possibilities are $a=\pm 12$ and $a=\pm 14$, and they don’t work.

- Show that 2S = S for all infinite sets
- CW complex structure on standard sphere identifying the south pole and north pole
- Arc length parameterization lying on a sphere
- A problem in the space $C$
- The rank and eigenvalues of the operator $T(M) = AM – MA$ on the space of matrices
- Quotient $M/M^2$ is finite dimensional over $R/M$ in local Noetherian ring?
- $T\circ T=0:V\rightarrow V \implies R(T) \subset N(T)$
- What are the prerequisites for taking introductory abstract algebra?
- $G$ a finite group such that $x^2 = e$ for each $x$ implies $G \cong \mathbb{Z}_2 \times … \times \mathbb{Z}_2$ ($n$ factors)
- Lemma vs. Theorem
- Integral of Bessel function multiplied with sine
- Forming Partial Fractions
- How do I solve this exponential equation? $5^{x}-4^{x}=3^{x}-2^{x}$
- Probability of balls in boxes
- Finding invertible elements in $\mathbb{Z}/m\mathbb{Z}$