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9h
comment Proof that if $a,b \in G$ and $a^4b = ba$ and $a^3 = e$ then $ab = ba$
To be generous to a terrible book proof, perhaps they were trying to do something analagous to the proofs involved with commuting higher powers of $a$ past $b$ - e.g., $a^3b=ba^3$ and $a^7=e$ implies $ab=ba$, as seen here ...
1d
comment Let $f:[-1,1]\to \mathbb{R}$ by $f(x)=x^4$. Determine the polynomial $p_2$ of degree less than or equal to 2 such that $||f-p_2||_2$ is minimal
What are $P_0, P_1, P_2$? Beyond that, you know that a degree-two polynomial (in whatever basis you choose) has three parameters (e.g., in the monomial basis it's just $ax^2+bx+c$ for some $a,b,c$); try writing out the $L_2$ norm of your difference explicitly in terms of $a,b,c$ and see if you can minimize that.
May
26
revised What is the difference between solving a second order linear homogeneous ODE using the Euler method and the method of undetermined coefficient?
tweaked the title to remove potentially-offensive term
May
25
comment How to find the greatest prime number that is smaller than $x$?
@DanaJ In addition to OP's presumably-practical question, there's also an interesting theoretical question here, and this answer takes a step in that direction. AFAIK it's not actually known that one can find the largest prime less than $x$ in time polynomial in the number of bits of $x$; the result holds with some reasonable assumptions (e.g. Cramér's conjecture), but that fact itself still needs the existence of a polynomial-time primality test like AKS.
May
25
comment Why do the vectors perpendicular to [1, 1, 1] and [1, 2, 3] fall on a line, as opposed to a plane?
What shape do the vectors perpendicular to only one of those vectors form? What does the intersection of two of those shapes look like?
May
25
comment polytope with 12 vertices and 48 edges
It certainly can't be a polyhedron, by Euler's formula. There would have to be 48-12+2=38 faces, but since each face has at least three edges there have to be at least 38*3/2=57 edges. I agree with user86418: why are you so confident such a polytope exists?
May
23
comment weird trig problem $\tan(\theta)=-\sqrt{2}\sin(\theta)$ on the interval $0 \leq \theta \leq 2\pi$
@john, when you multiply by $1/\sin\theta$, you introduce the possibility of additional solutions when $\sin\theta=0$. Basically, you can't just go from $ab=cb$ to conclude $a=c$; here $a=1/\cos\theta$ and $c=-\sqrt{2}$.
May
23
comment Given a fixed positive integer $k$, find the number of pairs of integers $(x,y) \in \mathbb{Z}^2$ such that $x^2+y^2=5^k$
OP is looking for the count of all such pairs - but with a little more information, this answer should be adaptable to give that as well.
May
23
comment Confused about transcendental numbers
You can use Vieta's formulas (among other ways) to prove this: if all of the roots of a monic polynomial are algebraic then all of the coefficients are algebraic.
May
23
comment Confused about transcendental numbers
@Ghassan that number is most likely transcendental (we don't actually know for sure!), but it's not 'super-transcendental' or anything like that.
May
22
comment Is $\mathbb{Q}(\pi) \cong \mathbb{Q}[[x]]/ \langle \sin(x) \rangle$?
@TheBluegrassMathematician The problem is that $\mathbb{Q}[x]$ already means something else, so your notation is innately confusing. Since there's a standardized notation for the object you're interested in, it's best to just use that notation.
May
22
comment Mandelbrot set of $c \cdot \cos(z)$
It's definitely not true that you can use the standard rules for the Mandelbrot set here: if $c=\frac32\pi$ then $f_c(0)=\frac32\pi\cos 0=\frac32\pi$ and then $f_c(f_c(0))=\frac32\pi\cos(\frac32\pi)=0$, so the origin enters a two-cycle. More generally, one can take arbitrary half-odd multiples of $\pi$ to see that any such bound must at least be dependent on $c$...
May
22
comment Conjecture: for even n without primitive roots modulo n, the set of $m \in Max(ord_n(k))$ contains one pair of primes $p_1+p_2=n$ (Goldbach)
2000 is still a very small number from the perspective of primality - you still have a good 15% of the numbers (30% of the odd numbers!) smaller than that being prime, so it's very hard to make judgements from the relatively meager data here. I would want to see numerical evidence up to at least $10^7$ or so and preferably $10^9$ before really putting much weight behind this.
May
21
awarded  Good Answer
May
21
comment Cauchy-Ramanujan Formula $ \displaystyle \sum_{\stackrel{m \in \mathbb{Z}}{m \neq 0}} \frac{\coth m \pi}{m^{4p+3}} $
What's the question here?
May
21
comment Variant Generating Function related to Euler Numbers
What was the source of the variant series, and is there any reason behind specifically the '2' in the '2x' factor? It seems unlikely that the coefficients of that series would have any particularly clean form...
May
21
awarded  Nice Answer
May
21
awarded  prime-numbers
May
20
answered How to force prime numbers into a line?
May
19
comment Identity element of a group
The specific mistake you made is that $e=ae$ doesn't necessarily imply that $a=1$ - you have to recognize that that involves a division by $e$ and so you need to consider what happens if you divide by zero (in this case, you find the intended solution).