john mangual
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 Apr14 comment What is $\left[\frac{1}{2}(p-1)\right]! \;(\text{mod } p)$ for $p = 4k+1$? @nayrb A similar statement is Gauss' Lemma in Number Theory which counts the number of residues of $\{ a, 2a, \dots, \tfrac{p-1}{2} a\}$ less than $p/2$. Apr14 comment What is $\left[\frac{1}{2}(p-1)\right]! \;(\text{mod } p)$ for $p = 4k+1$? @nayrb the original theorem just asks about whether $\nu$ is even or odd. that might be good enough. Apr14 comment What is $\left[\frac{1}{2}(p-1)\right]! \;(\text{mod } p)$ for $p = 4k+1$? whoops :-) my 2nd question would be nontrivial if I said "less than $\frac{p}{4}$" Apr8 comment Fundamental unit in the ring of integers $\mathbb Z[\frac{1+\sqrt{141}}{2}]$ Another version of this discussion appears in Section 2.7 of Borevich + Shafarevich number theory book (PDF) in the Chapter on "Decomposable Forms", meaning that $x^2 - dy^2 = (x - \sqrt{d} y)(x + \sqrt{d}y)$ has complete factorization. Pell's equation can be thought of as a special case of Dirichlet Unit Theorem in Section 2.4 or part of Quadratic Forms in Section 2.7 Apr8 comment How find the postive $m,n$,such $a^n\equiv 1\pmod m$ Consider mod 8. $\phi(8) = 4$ so $a^4 \equiv 1 \mod 8$ whenever $(a,8)=1$ - in fact $a^2 \equiv 1 \mod 8$. The group structure of $\mathbb{Z}/m\mathbb{Z}$ is tied to the prime factorization of the units $\phi(m)$, which depends on the prime factorization of $m$. Apr7 comment Fundamental unit in the ring of integers $\mathbb Z[\frac{1+\sqrt{141}}{2}]$ The Pigeonhole Principle and this solution of Pell's equation is due to Dirichlet Apr7 comment Fundamental unit in the ring of integers $\mathbb Z[\frac{1+\sqrt{141}}{2}]$ math.uzh.ch/index.php?file&key1=31685 Mar18 comment Official name of Fermat's $x^2+3y^2$ theorem? @Ark do you just want the name or the proof? Mar18 comment Official name of Fermat's $x^2+3y^2$ theorem? amazon.com/Primes-Form-Fermat-Complex-Multiplication/dp/… Mar16 comment continued fraction of the roots of $x^2 - \frac{53793390359}{1088391168}x + \frac{823543}{12230590464} = 0$ Gee these numbers are causing a stack overflow on your computer. Perhaps it's safe to say with such large numbers, the continued fraction digit is essentially random and that an exact answer is beyond reach. Mar14 comment Geometric interpretation for sum of fourth powers @alex.jordan This problem is very hard. I have no idea where these sum-of-powers formulas come from. While Erhart polynomials generalize figurate numbers of any kind, I am learning they are difficult to compute in general. They are related to Pick's theorem or Dehn-Somerville relations. Mar12 comment Geometric interpretation for sum of fourth powers @alex.jordan Did you notice the sum of squares divides sum of 4th powers? $$\sum k^2 =\frac{1}{6}n(n+1)(2n+1) \Bigg| \sum k^4 = \frac{1}{30}n(n+1)(2n+1)(3n^2+3n-1)$$ The blog you mentioned has two solutions. One uses a triangular lattice. $$.$$ The volume of a pyramid is $\frac{1}{3}Ah$ where $A$ is the base and $h$ is the height. By Cavalieri principle we don't worry if the Pyramid is slanted or not. Mar12 comment Geometric interpretation for sum of fourth powers Here is a cute article on Faulhaber's triangle Mar12 comment continued fraction of the roots of $x^2 - \frac{53793390359}{1088391168}x + \frac{823543}{12230590464} = 0$ @MichaelBurr No I have not! Even though Vincent's theorem is the basis of the fastest method for the isolation of the real roots of polynomials, it was almost totally forgotten, having been overshadowed by Sturm's theorem Mar8 comment continued fraction of $3 + 17\sqrt{3}$ how did you know the algorithm terminated? Mar7 comment continued fraction of $3 + 17\sqrt{3}$ @Amzoti I took code for the GCD funciton from StackOverflow and modified it to handle algebraic numbers in $\mathbb{Z}[\sqrt{d}]$. stackoverflow.com/questions/11175131/… Feb28 comment Fast search of local positive quadruples on the sphere cs.stackexchange.com since you are asking about runtime Feb28 comment Efficiently producing certain kinds of examples of the application of Euclid's algorithm @MichaelHardy I am working on it :-) Notice gcd = 1 with probability $\frac{6}{\pi^2} \approx \frac{2}{3}$ for two random numbers! Unfortunately, these may have large factors, so they don't follow your smoothness condition. Feb27 comment Identity with nested sum taken over divisors of $\gcd$'s @MarkusScheuer you are right. this is a placeholder for a more complete solution. I think Sary has the right approach but s/he struggles with the Dirichlet series a bit. Symmetry will lead to the right answer. Feb23 comment duality theory question In your first equation, $x^T x = ||x||^2$ the norm of the vector. So you ask for the point of minimum norm across a certain affine subspace. Then you can generalize the point-to-line distance formula.