ShreevatsaR
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28,010
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 Jul 5 revised Matsunaga's Method for solving $x^2+y^2=p$ you only *really* understand something when you can write code for it. :-) Jul 5 revised Matsunaga's Method for solving $x^2+y^2=p$ slightly better, will deal with the remainders later Jul 4 revised Matsunaga's Method for solving $x^2+y^2=p$ get rid of the deltas, not sure if it's better Jul 4 revised Matsunaga's Method for solving $x^2+y^2=p$ rewrite Jul 4 revised Matsunaga's Method for solving $x^2+y^2=p$ all the examples Jul 4 revised Matsunaga's Method for solving $x^2+y^2=p$ another example Jul 4 answered Matsunaga's Method for solving $x^2+y^2=p$ Jul 2 comment Roulette betting system probability @Theo: Your question was, "what is the probability of winning at least one bet of the 12". The above is the answer, for any (feasible) betting system. Now you seem to be asking newer (and very different) questions, such as what if the betting system is itself infeasible, and what the probability of winning some money is. Please be clear about what your question(s) is/are. Jul 2 answered Roulette betting system probability Jul 2 comment Roulette betting system probability For those without knowledge of roulette: Is it correct that the probability of winning a particular bet is $\frac{18}{38}$? Or is the winning probability something else? Also the payoff: if you bet an amount $x$ and you win, you gain another $x$, while if you lose the bet, you lose your $x$. Is that correct? Jul 2 comment Jensen inequality No. $\phantom{a}$ Jun 27 revised Can $x^{x^{x^x}}$ be a rational number? undo needless change Jun 27 reviewed Reject Can $x^{x^{x^x}}$ be a rational number? Jun 27 comment How do I understand the meaning of the phrase “up to~” in mathematics? Good point. As an example of sloppy thinking I offer myself: if asked to make the statement more explicit without "up to", I'd probably have written something like "If $r$ and $s$ are positive integers, and $p_1,\dots,p_r$ and $q_1,\dots,q_s$ are prime numbers satisfying $p_1\le\dots\le p_r$ and $q_1\le\dots\le q_s$, then $r=s$ and $p_i=q_i$ for all $1\le i\le r$." Now I think of it, this statement, while correct, is actually weaker than the original statement, and has essentially pushed the issue of order under the rug. (This picks one particular bijection $\sigma$ as above of course.) Jun 25 comment What is the summation notation for the Fibonacci numbers? $F_n = \sum_{i = 1}^{F_n} 1$ ? :-) Jun 25 comment Questions on “All Horse are the Same Color” Proof by Complete Induction Very clear explanation of the part that was confusing to the OP -- good work understanding what the OP's confusion was! Jun 25 comment Finding Big-O with Fractions @BustedZen: Yes, for the first function, we can see that $f(x)=O(x)$ because as you say, just the $x$ term will have the greatest impact on the value of $f(x)$ (the other terms become very small in comparison). But we have to prove that it is $O(x)$ — what is the definition of $O(x)$ that you know? The definition I used is that $f(x)=O(x^n)$ if there exists a constant $c$ such that for all sufficiently large $x$, we have \$f(x)