# Tag Info

• 118k

### The largest digital sum of the square of an n-digit number

After 10 + days calculating, I got the following sequence : {13, 31, 46, 63, 81, 97, 112, 130, 148, 162, 180, 193, 211, 229, 244, 262, 277, 297, 310, 331, 343, 360, 378, 396, 406, 423, 436, 454, 469, ...
• 31

• 30.6k

### Six consecutive positive integers with certain shape

With the analysis from Stanley, Joachim and Max Alekseyev, maybe I can solve my question now. Because two different integers with the form 4k+2 can not appear in these six numbers, when modulo 4, they ...

### Six consecutive positive integers with certain shape

This is just to show that in sextuples of interest $2x^2 - 3y^2 = -1$, while difference $2$ is not possible. First note that neither $|2x^2-2y^2|<6$ nor $|3x^2-3y^2|<6$ is soluble in distinct ...
• 30.6k

### Six consecutive positive integers with certain shape

I'd like to add something from the viewpoint of heuristics. The most restrictive condition in this problem seems to be "occurence of a number of $p$-adic valuation $1$ for a small prime $p$" ...
• 2,643
Accepted

### Ask for a generating function or an explicit expression of a triangle of positive integers

The generating function: $${\cal C}(x,y) = \sum_{n,k\geq 0} C_{n,k} x^n y^{2k}$$ has the following explicit form: $${\cal C}(x,y) = \frac{\arctan(y)}{y(1-x(1+y^2))}.$$ For "one more problem",...
• 30.6k
Given your question, it seems that one should study the pair of equations $\displaystyle 2x^2 - 3y^2 = -1 \quad \text{and} \quad 2x^2 - 3y^2 = 2.$ (In fact, the quadratic field $\mathbb{Q}(\sqrt{6})$ ...