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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 $f(x)<cx^n$. (Picked a constant on the other side as well, to show that $f(x)$ is actually $\Theta(x)$, i.e. it is not $O(\text{something asymptotically smaller than }x)$.)
Jun
24
comment Is there always a prime $p$ so that the largest prime factor of $p^2+i$ not exceeding $p$ for $-k\leq i \leq k$?
In other words, given $k$, you want $p$ such that the $2k+1$ numbers around $p^2$, namely $p^2 - k$ to $p^2 + k$, are all made up only of prime factors at most $p$. Is this reading correct?
Jun
24
comment Mathematical Quine
+1, but Tupper's "self-referential" formula (which he never named so, in the excellent paper where he used it as example) just takes a bitstring (the region) and forms a bitmap out of it. This is like a program that echoes its input: sure, when given its own code as input it does print it out, but that doesn't make it a quine. However, Jakub Trávník has written a true mathematical "quine", whose graph contains all the information necessary to recreate it: here.
Jun
24
answered Finding Big-O with Fractions