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 Nov 3 comment Modular Arithmetic - Find the Square Root Perhaps “This gives us 2⋅2=4 primes” should say “This gives us 2⋅2=4 roots”. Also fix “We can firs calculate” Aug 5 comment The largest possible prime gap? @BradGraham, it isn't (eg 2*3*5-1 = 29) so I have edited to correct that error. Thanks. Nov 21 comment Is the function $f( x)=1/|x|^{1/2}$ Lipschitz continuous? The stated $x_i$ don't work; eg if C = 1+e for small positive e, $|f(x_1)-f(x_2)| < C|x_1-x_2|$. However, $x_1 = \frac{1}{C^2}$ and $x_2 = \frac{1}{2C^2}$ work. Nov 15 comment Percentage of primes among the natural numbers @MartinSleziak Moose can un-accept one answer and accept a different one at any time, should he or she so desire. Nov 15 comment Problem on Divide-and-Conquer Relation Although you say "Each player who loses in the first round gets nothing", the problem says "a player who loses in the first round gets $100". Nov 15 comment Counting walks on colored graphs Can you explain your$3*2^{k−1}$walks on k 3-colored vertices formula? At least at k=1 and k=2 the numbers appear to be twice as large; e.g {br, bg, gr, gb, rg, rb} at k=1, and {gbg, grg, gbr, grb, etc} at k=2. Nov 14 comment Algorithms and Simulation Were you given a definition of "optimal"? Or any limitations? (I.e. for (b) it looks like$\sigma^2=1$should be optimal for the obvious def of optimal) Oct 12 comment Find the minimum number of links to remove from digraph to make it acyclic An acyclic graph contains no cycles at all, rather than merely a vertex that's not in a cycle. Oct 11 comment Powers as a complete residue system modulo$p$? Also see MSE question re the number of equivalence classes of (in notation of present question)$k^a\mod p$Oct 10 comment How to express an irrational as a continued fraction in computer with high precision? Rather than "up to 16 bits" being not quite right, it's totally wrong. Typical IEEE754 64-bit double-precision numbers use effectively 53 bits for mantissa, giving ~ 16 decimal digits accuracy. (15.95... = 53 * ln(2)/ln(10).) Sep 19 comment Maximizing growth rate in betting on multiple events Question 1: Suppose$b_{134}$is the the fraction of stake$s$that is bet on$\{1,3,4\}$. Suppose$t=b_{134}*s/3$. If events 1 and 3 win and event 4 loses, is the payoff equal to$(1+c_1+1+c_3)*t$(that is, the amount$s*b_i$spreads evenly across the elements of bet #$i$) or is it some other value? Question 2: Do all events in a set have to win to get a payoff? Question 3: Since all bets are simultaneous, there appears to be no "long run" to be concerned about, and no reason to act other than Charles suggested in his answer, right?. Sep 19 comment Maximizing growth rate in betting on multiple events Ok, let an$i$be given. What events are elements of bet #$i$? Is the amount$b_i$spread evenly across the elements of bet #$i$? Does bet #$i$get paid off and its result reported to you before you make bet #$i+1$? Sep 19 comment Maximizing growth rate in betting on multiple events Is one allowed to bet on the same alternative multiple times? Is the order of bet payoffs specified? Is$b_i$an amount bet on event$i$, or on some subset of the events? Sep 10 comment Conditions on polygon to ensure equal interior angles or opposite sides Yes, the negation of (c) is (c'), "not all opposite sides have equal lengths". Your comment re "if opposite sides are parallel then opposite angles are equal" becomes obviously true via another comment's suggestion to "draw the diagonal" (i.e. call upon Euclid's Proposition 29, "A straight line falling on parallel straight lines makes alternate angles equal to one another..."). Aug 22 comment Number of$k^p \bmod q$classes when$q\%p > 1$@awllower, the "q%p>1" part of the title was used as shorthand for "q != 1 mod p". The "number of classes" part refers to the number of equivalence classes, mod q, of k^p. How would you phrase it? Aug 22 comment Number of$k^p \bmod q$classes when$q\%p > 1$thanks for the answer, @awllower. I agree with what you say, except that the theorem's last assertion is stronger than what I wanted to show -- it also characterizes the number of classes when q == 1 mod p. For my purposes (given p, finding q so that k^p falls into q-1 classes, so that computed numbers can quickly/efficiently be recognized as probable p'th powers) I don't care about that case, although other people could very well care about it. Aug 15 comment Is it possible to efficiently factor a semiprime given a bit-permutation relating the factors? @Charles, you are right, I apologize for that. I'm now going to edit the "n=#bits" references in my answer to "u=#bits" to reduce longer-term confusion. Aug 15 comment Is it possible to efficiently factor a semiprime given a bit-permutation relating the factors? Perhaps you wrote$n$rather than$p$in the above.$p$is the prime being factored,$n$is the number of bits in its binary representation. But your point is correct --$O(1)$is too pessimistic. If we suppose a modern method is$O(p^{1/3})$and solve$p^{1/3} = 3^k$, we get$k\approx n\cdot{\ln2\over3\ln3}\approx0.21n$; so brute force is competitive for these numbers if no more than 42% of the bits change places. Aug 14 comment Winning on Prize Bonds @francis, the "squiggly equals" stands for approximately equal. The "upside down y" is a greek lambda, as commonly used for the parameter of a Poisson distribution. Re Binomial distribution and approximations to it, the Wikipedia page I linked to explains in detail; briefly, a Binomial dist. expresses the probability of getting k out of n flips to come up heads, with a given probability p for a head. Re "probability of success on one draw", I approximated a binomial by a geometric-probability ratio; the other answers don't make that approximation. Aug 13 comment Number of$k^p \bmod q$classes when$q\%p > 1\$ @Jyrki, I'd like to think I was conversant with the techniques you mentioned, when I took number theory classes long ago. However, I've forgotten quite a lot. The straightforward and basic techniques of André's answer are more helpful to me, while I expect practicing mathematicians will prefer the conciseness of your hints.