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 Nov 18 answered Proving $n^3$ is even iff $n$ is even Nov 18 comment D.w. $p_i>\sigma(p_1^{a_1}p_2^{a_2}…p_{i-1}^{a_{i-1}})\forall i \in [1,\omega(n)]\iff d_j>d_1+d_2+…+d_{j-1} \forall j \in [1,\sigma_0(n)]$ I have not worked through it all rigorously, but I note that if (1) holds then the divisors, in order, will run through the powers of the smallest prime uninterrupted, then the same sequence with a factor of the second prime included, and so on through all the combinations of the first two primes, then the pattern will repeat with the third prime, and so on. Obviously the divisor which is the second prime will meet (2) as this is the same condition as (1) - the same for the third, etc. prime. A quick survey looks like the other divisors will not have any more restrictive conditions. Nov 16 comment Why can't you square both sides of an equation? I think this best answers the updated question. Taking this approach, I would also show operations like $0 \cdot f(x)$ that were given in a different answer to drive home the idea that it is not just squares that are not reversible; hopefully the students will remember the more general principle rather than the specific instance. Nov 16 comment sum of all coprimes of a number. The difficulty is not in counting how many coprime numbers there are (read about the $\phi$ function mentioned), but in determining which values they have in order to sum them up. By showing there is a symmetry to them you can determine their average value without knowing exactly which numbers they are. Nov 15 comment D.w. $p_i>\sigma(p_1^{a_1}p_2^{a_2}…p_{i-1}^{a_{i-1}})\forall i \in [1,\omega(n)]\iff d_j>d_1+d_2+…+d_{j-1} \forall j \in [1,\sigma_0(n)]$ I can't say for sure since I only looked at certain patterns of prime signatures. But it looks like it will hold for all numbers with exactly two distinct primes. I didn't find any cases in the patterns I looked at where (2) became more restrictive than (1). Nov 15 answered D.w. $p_i>\sigma(p_1^{a_1}p_2^{a_2}…p_{i-1}^{a_{i-1}})\forall i \in [1,\omega(n)]\iff d_j>d_1+d_2+…+d_{j-1} \forall j \in [1,\sigma_0(n)]$ Nov 11 comment Do I influence myself more than my neighbors? This matrix is not symmetric, which was stated as a condition in the question. Can you make your point with a symmetric matrix? Apr 14 comment How to solve $\log(x)=\frac{\log(1,04)}{6}$ without calculator? Can you provide more detail on what you are studying? Are you expected to use printed log tables? Are you expected to express the answer as a number or as an expression? Mar 27 comment compute minimum distance between point and great arc on sphere You can compute bearing by forming the triangle with $Q$, $P$, and the north pole; then the bearing is one of the angles of the triangle. The triangle of two points and the pole is used frequently in this way. As I said before, though, be careful of $\acos$ of small angles - they aren't very accurate, and you can solve an alternate equation to avoid the problem. If you're not familiar with the spherical trig equations, you can start with Wikipedia: en.wikipedia.org/wiki/Spherical_trigonometry Mar 26 comment compute minimum distance between point and great arc on sphere No, bearing for an arbitrary point is not the same as any of the rotation angles. It is a direction, relative to the line of longitude at that location. As far as degrees of arc, that means "the length of an arc on the surface of the sphere whose endpoint position vectors form an angle of x degrees at the center of the sphere". Mar 22 comment compute minimum distance between point and great arc on sphere In navigation, bearing is the angle from North (or the angle made with a line of longitude). I avoided reference to North since I assumed your great circle could have any orientation. Unfortunately, the coordinates and there reference points are generally different between navigation, mathematics, and the use of spheres for viewpoint calculation. I did take care to say "degrees of arc" for distances, so all the other places I should have been referring to angles on the surface of the sphere. I haven't yet looked up up your application; if I did I might have a better terminology for you. Mar 22 comment compute minimum distance between point and great arc on sphere I did leave quite a bit to still be worked through. One way to find the "pole" of the great circle would be to compute the bearing from your $Q$ to $R$, add 90 degrees and find the point that is 90 degrees of arc away, since the pole is always 90 degrees of arc away. Note that I don't know your application - in some of these operations you need to be careful of loss of precision when choosing which formulae to use to get each quantity. Mar 22 comment How to calculate the number of pieces in the border of a puzzle? I really want to vote this up, but can't in good conscience since it's not (meant to be) helpful. :-) The other answers aren't all that helpful either, due to the issues raised in the comment. Mar 22 answered What is the probability you guess the number I am thinking of? Mar 22 comment Simplifying Catalan number recurrence relation Typo in the second line: first $(n-1)$ is $(n+1)$. But I can't do one-character edits. Mar 22 comment compute minimum distance between point and great arc on sphere I didn't mention this because I don't know your level of familiarity with sperical trig, but one way to see if the tangent point is on the arc is to compare the bearings from the pole to each end point and the tangent point. Though you may need to be careful regarding which quadrant your results are coming back in; i.e., 0 is between 330 and 30. Mar 22 comment compute minimum distance between point and great arc on sphere Actually, I've found that most GIS folks aren't aware of these complexities - they deal with too little of the earth at a time, or depend on the libraries to do it for them. I actually know it from real-world orbital trajectory calculations, so here or physics would be the right places. Mar 21 comment what is the ratio of their speed? That should be 280-40 = 240, not 220 Mar 21 answered What are some ways to find the minimum of an expression? Mar 21 revised Factorials and Arithmetic Progression. fix spelling and typos