All Questions

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The cartesian product $\mathbb{N} \times \mathbb{N}$ is countable

I'm examining a proof I have read that claims to show that the Cartesian product $\mathbb{N} \times \mathbb{N}$ is countable, and as part of this proof, I am looking to show that the given map is ...
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What is the term for a factorial type operation, but with summation instead of products?

(Pardon if this seems a bit beginner, this is my first post in math - trying to improve my knowledge while tackling Project Euler problems) I'm aware of Sigma notation, but is there a function/name ...
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In classical logic, why is$(p\Rightarrow q)$ True if both p and q are False?

I am studying entailment in classical first-order logic. The Truth Table we have been presented with for the statement $(p \Rightarrow q)\;$ (a.k.a. '$p$ implies $q$') is: ...
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Taking Seats on a Plane

This is a neat little problem that I was discussing today with my lab group out at lunch. Not particularly difficult but interesting implications nonetheless Imagine there are a 100 people in line to ...
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Comparing $\pi^{e}$ and $e^{\pi}$

How can I calculate without calculator or something like this the values of $\pi^{e}$ and $e^{\pi}$ in order to compare them ?
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Is there possibly a largest prime number?

Prime numbers are numbers with no factors other than one and itself. Factors of a number are always lower or equal to than a given number; so, the larger the number is, the larger the pool of ...
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Making Change for a Dollar (and other number partitioning problems)

I was trying to solve a problem similar to the "how many ways are there to make change for a dollar" problem. I ran across a site that said I could use a generating function similar to the one quoted ...
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Is $\lim\limits_{k\to\infty}\sum\limits_{n=k+1}^{2k}{\frac{1}{n}} = 0$?

Is $\lim\limits_{k\to\infty}\sum\limits_{n=k+1}^{2k}{\frac{1}{n}} = 0$ I.e - does the second half of the harmonic series go to zero? I know that for a finite number of terms the limit of the sum is ...
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Finite Sum of Power?

Sorry for the remedial math but: Can someone tell me how to get a closed form for $$\sum_{k=1}^n k^x$$ For $x = 1$, it's just the classic $n(n+1)/2$. What is it for $x > 1$?
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$|G|>2$ implies $G$ has non trivial automorphism

Well, this is an exercise problem from Herstein which sounds difficult: How does one prove that if $|G|>2$, then $G$ has non-trivial automorphism? The only thing, i know which connects a group ...
984 views

Prove: $\int_0^1 \frac{\ln x }{x-1} d x=\sum_1^\infty \frac{1}{n^2}$

I'd like your help with proving that $$\int_0^1 \frac{\ln x }{x-1}d x=\sum_{n=1}^\infty \frac{1}{n^2}.$$ I tried to use Fourier series, or to use a power series and integrate it twice but it didn't ...
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Prove that if $\gcd( a, b ) = 1$ then $\gcd( ac, b ) = \gcd( c, b )$

I know it might be too easy for you guys here. I'm practicing some problems in the textbook, but this one really drove me crazy. From $\gcd( a, b ) = 1$, I have $ax + by = 1$, where should I go from ...
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Union of two vector subspaces not a subspace?

I'm having a difficult time understanding this statement. Can someone please explain with a concrete example?
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Number of monic irreducible polynomials of degree $p$ over finite fields

Suppose $F$ is a field s.t $\left|F\right|=q$. Take $p$ to be some prime. How many monic irreducible polynomials of degree $p$ can exist over $F$? Thanks!
How do I prove that $x^p-x+a$ is irreducible in a field with $p$ elements when $a\neq 0$?
How do I prove that $x^p-x+a$ is irreducible in a field with $p$ elements when $a\neq 0$? Right now I'm able to prove that it has no roots and that it is separable, but I have not a clue as to ...