john mangual
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 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 revised Fundamental unit in the ring of integers $\mathbb Z[\frac{1+\sqrt{141}}{2}]$ added 827 characters in body Apr7 answered Fundamental unit in the ring of integers $\mathbb Z[\frac{1+\sqrt{141}}{2}]$ Apr7 comment Fundamental unit in the ring of integers $\mathbb Z[\frac{1+\sqrt{141}}{2}]$ math.uzh.ch/index.php?file&key1=31685 Mar28 asked formula for the number of perfect squares mod $N$ Mar23 revised Extending a Chebyshev-polynomial determinant identity after a long time I decided my previous solution was wrong. Mar23 revised Extending a Chebyshev-polynomial determinant identity added 482 characters in body Mar23 revised Extending a Chebyshev-polynomial determinant identity added 482 characters in body Mar23 answered Extending a Chebyshev-polynomial determinant identity 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/… Mar18 asked proving Fermat's theorem on $p = x^2 + 3y^2$ Mar18 revised Is $\sum_{k=1}^{m-1}\frac{1}{\sin^2\frac{k\pi}{m}}=\frac{m^2-1}{3}$ true for $m\in\mathbb N$? deleted 205 characters in body 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 revised Geometric interpretation for sum of fourth powers added 440 characters in body Mar12 comment Geometric interpretation for sum of fourth powers Here is a cute article on Faulhaber's triangle