Reputation
7,783
Top tag
Next privilege 10,000 Rep.
Access moderator tools
Badges
9 26
Newest
 Civic Duty
Impact
~66k people reached

Apr
13
awarded  Necromancer
Apr
8
revised How to prove Chebyshev's result: $\sum_{p\leq n} \frac{\log p}{p} \sim\log n $ as $n\to\infty$?
defective link
Apr
8
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
Apr
8
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 $m$.
Apr
7
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
Apr
7
revised Fundamental unit in the ring of integers $\mathbb Z[\frac{1+\sqrt{141}}{2}]$
added 827 characters in body
Apr
7
answered Fundamental unit in the ring of integers $\mathbb Z[\frac{1+\sqrt{141}}{2}]$
Apr
7
comment Fundamental unit in the ring of integers $\mathbb Z[\frac{1+\sqrt{141}}{2}]$
math.uzh.ch/index.php?file&key1=31685
Mar
28
asked formula for the number of perfect squares mod $N$
Mar
23
revised Extending a Chebyshev-polynomial determinant identity
after a long time I decided my previous solution was wrong.
Mar
23
revised Extending a Chebyshev-polynomial determinant identity
added 482 characters in body
Mar
23
revised Extending a Chebyshev-polynomial determinant identity
added 482 characters in body
Mar
23
answered Extending a Chebyshev-polynomial determinant identity
Mar
18
comment Official name of Fermat's $x^2+3y^2$ theorem?
@Ark do you just want the name or the proof?
Mar
18
comment Official name of Fermat's $x^2+3y^2$ theorem?
amazon.com/Primes-Form-Fermat-Complex-Multiplication/dp/…
Mar
18
asked proving Fermat's theorem on $p = x^2 + 3y^2$
Mar
18
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
Mar
16
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.
Mar
14
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.
Mar
12
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.