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If you want, you can use Lambert's W function. Let $z:=x+4$, so that the equation becomes $3^{z-4}=z$, so that $-3^{-4}=-z3^{-z}$, which upon inverting with Lambert's $W$ gives: $x=-\frac{W(-\ln(3)3^ …

Let $D(f,s):=\sum_{n=1}^\infty \frac{f(n)}{n^s}$, otherwise known as a Dirichlet series. When $f$ is a multiplicative, number theoretic function, $D(f,s)$ tends to be expressed as a rational product o …

This is not yet an answer, but in terms of $B$ (beta) functions, one has $B(a,1-a-b)+B(b,1-a-b)+B(a,b)=\displaystyle\frac{B\left(\frac{b}{2},\frac{1}{2}\right)}{B\left(\frac{1-a}{2},\frac{a+b}{2}\rig …

The word intuitive by definition means that something "feels correct", and is very myopic. The fact that there are so many different proofs is in itself a gift because you can pick your favorite and i …

If you want something more on the expository side, "An Invitation to Modern Number Theory" by Miller and Takloo-Bighash builds up both L-functions and random matrices from the ground up, later connec …

Let $c(n)$ be the number of Self avoiding walks (SAW) of length $n$ on an infinite lattice $L$. Are there any known non-geometric interpretations of $c(n)$?. For example, is there a number theoretic v …

Writing up the comment: You just need to "pixelate" the line by finding all lattice boxes that it crosses: Then the answer vector $v$ must connect to one of the corners of the shaded boxes. Instead o …

Introduction: Let A be a subset of the naturals such that $\sum_{n\in A}\frac{1}{n}=\infty$. The Erdos Conjecture states that A must have arithmetic progressions of arbitrary length. Question: I w …