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 Dec 25 comment On “familiarity” (or How to avoid “going down the Math Rabbit Hole”?) I too have had difficulty getting any traction on new areas from sources such as Wikipedia, and moreover am frustrated to see areas I do know explained in ways that will not help a beginner. For an autodidact, sources of free online textbooks are valuable; reddit.com/r/mathbooks is one place to start. Dec 14 comment $x$ values for $2x + |x| = 3 + |x+1|$ It is OK to leave work for the reader, but you need to state you have not completed the problem lest the OP think you have demonstrated a complete solution. Dec 13 comment Determinate a continuous function given a few coordinates. I also like rational polynomials for practical work. They keep the degree of the polynomial down so they are easier to invert, and often the singularity is outside the region of interest, or is a feature of the function you are fitting to. Dec 13 comment Determinate a continuous function given a few coordinates. Write a general polynomial of degree one less than the number of points, then create a system of n equations which are the polynomial evaluated at each point. Then solve the system of equations to get the polynomial coefficients. Dec 10 comment Values of a, b, and c that the curve $y = ax^3 + 3x^2 + bx + cx + e^x$ has one point of inflection? Given that $e^x$ is positive will drive you to a negative x. Since it won't be too small you can probably use an estimate for $e^x$ around that x. Dec 2 comment Does every record of the arithmetic derivative of natural numbers occur at a practical number? For a moment I thought they might be products of primorials (oeis.org/A025487) which is a tighter restriction, but 640 is the first one that is not. I'll have to keep thinking about it. Nov 28 comment Trigonometry triangles Did you use c = - 17,500? I get two solutions, around F = 67 and F = - 260, which both work in the original equation. Nov 28 comment Can we determine which statements are incomplete due to Godel? Thanks for the clarification. Nov 28 comment Can we determine which statements are incomplete due to Godel? So I understand your answer as: Yes, it is possible that any of these conjectures are true but unprovable, and No, we cannot know which questions are the unprovable ones. Nov 28 comment Lower bound for logarithm? Since we were discussing bounds A "close to" 0, I assumed we were only interested in a bounding function for $x << 1$. Nov 26 comment Prove that every practical number is either a power of two or a power of two times a non-trivial polygonal number @PeterTaylor I will do it if you wish. I want more experience updating OEIS. Let me know. p.s. love the domain name, and those laser swords. Nov 24 comment Positively non-positive (from Brilliant.org) Whats wrong with my method? @YanYau, note that although a single case is not sufficient to prove that an N works, a single counter-example is sufficient to prove that a given N doesn't work, since you were asked to prove a condition always holds. Nov 24 comment Derivative word problem - $s''(t) = a$, i.e the acceleration is constant and $(1) [s'(t)]^2 = 2as(t)$ Note that the prior question asks you to show why the solution is not in that form, so you shouldn't be surprised that there is no 'c' in the new form you are given. Nov 24 comment Proving for every odd number $x$, $x^2$ is always congruent to $1$ or $9$ modulo $24$ I think you should make it its own answer, as this seems to be the real issue with the OP's understanding, not squaring and reducing numbers. Nov 24 comment Number of ways to arrange $5$ a and $5$ b letter As Andre says, how many places can the first 'a' go? The second? ... The fifth? But the 'a's are indistinguishable, so for each pattern we've produced, how many times did we create it by placing the 'a's in a different order? The end result will be what Adi wrote. Nov 24 comment Is $x^2$ always congruent to $(y-x)^2$ modulo $y$? How could you prove the cases where its true? that should be $-2xy$ in the expansion, luckily it doesn't change the conclusion. Nov 23 comment Prove that every practical number is either a power of two or a power of two times a non-trivial polygonal number @Jaycob, for the moment I was looking at lines of practical numbers for $x^2 \pm y$ or $P(x) \pm y$ (P being the pronic numbers), in a parallel to the patterns seen for primes in the Sacks number spiral. BTW, it was answering your other question that started me on the practical numbers and I haven't been able to give them up yet. Nov 23 comment Twin Prime Powers To the original question, since powers of three are so rare it is probably quickest to simply enumerate and test them, as I presume @MatthewConroy has done already up to $3^{41}$. Nov 23 comment Twin Prime Powers I will note for the archive: the reason one number must be a power of $3$ is that all primes, and prime powers, $\gt 3$ are $\pm 1 \pmod 6$, therefore to extend a progression to three terms, one must be $3 \pmod 6$ which, being divisible by three and a prime power, can only be a power of three. Nov 22 comment Limit of $\left(1-\frac{1}{n^2}\right)^n$ Thanks for catching that. I was so focused on the MathJax formatting that I forgot to include the sign.