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 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. Nov 22 revised Limit of $\left(1-\frac{1}{n^2}\right)^n$ fixed missing minus signs Nov 22 comment Speeding Past a Car Just to be pedantic, this is an example of a poorly written word problem. The same "rate of traffic flow" does not mean that all the cars are even spaced on the highway, just that in aggregate there are the same number. But, as this is a homework problem this is not an issue with the OP, but with his or her teacher or textbook. It just bothers me when students have to "guess what the question wants" because this is the sort of thing that will get you in trouble with real world problems. Nov 22 answered Limit of $\left(1-\frac{1}{n^2}\right)^n$ Nov 21 comment Positively non-positive (from Brilliant.org) Whats wrong with my method? OK, I see that was sufficient to answer the original post ("Why is that not correct") but I can't help trying to determine what the right answer should be :) Nov 21 comment Positively non-positive (from Brilliant.org) Whats wrong with my method? I agree with your point, but isn't the sum of products for your example $(N-2)+(1-N)+(1-N) = -N \le 0$? In which case it is not a counterexample. Nov 21 comment How do I find the sum of prime factors of $(1750 + 1225)^{1229}$? @bert, is there a typo in the problem statement? In your comment you are using $1129$ and $2258 = 2 \cdot 1129$, but the question has $1229$ - not that it changes the principles though. Nov 21 comment How do I find the sum of prime factors of $(1750 + 1225)^{1229}$? I think you have an arithmetic error. $5^3 \cdot 41$ is $5125$; $2975$ is $5^2 \cdot 7 \cdot 17$. Nov 19 comment Prove that every practical number is either a power of two or a power of two times a non-trivial polygonal number @Peter, I hope you will reconsider and submit a link to OEIS. They are willing to provide links to more extensive lists - there are links to lists of 100,000 primes, and 1200 highly composite numbers. I for one would like more practical numbers to use for pattern analysis. Nov 18 comment For every integer $n$, the remainder when $n^4$ is divided by $8$ is either $0$ or $1$. You seem to be there. What is the remainder of $8m$ (or $8m+1$) when divided by 8? Nov 18 comment Show the series $\sum_{j=1}^{\infty} \frac{(2^j)+ j}{(3^j) - j}$ converges your original answer would have been much more helpful if you had named which principle you were applying rather than being so terse. I did remove a downvote though. Nov 18 comment Show the series $\sum_{j=1}^{\infty} \frac{(2^j)+ j}{(3^j) - j}$ converges As the original poster already noted, this sequence is smaller than the original, not larger, so it is insufficient to show convergence. Nov 18 comment Can't isolate $x$ for this equation Don't get so caught up in the algebra that you forget to check the values at the ends of the interval, assuming you were given an interval less than infinity. Nov 18 comment Proving $n^3$ is even iff $n$ is even Also a very nice way to approach this proof. Nov 18 revised Prove that one of the numbers k,k+1, . . . ,k+(n-1) is divisible by n. Changed k to n assuming the problem is not as trivial as written Nov 18 suggested approved edit on Prove that one of the numbers k,k+1, . . . ,k+(n-1) is divisible by n. Nov 18 revised Proving $n^3$ is even iff $n$ is even add relationship to prime factorization theorem