I am trying to proof $ab = gcd(a,b)lcm(a,b)$. The definition of lcm(a,b) is as follows: t is the lowest common multiple of a and b if it satisfies the following: i)a | t and b | t ii)If a | c and b | c, then t | c. Similiarly for the gcd(a,b). Here is my proof: Case I: gcd(a,b) $\neq$ 1 ...

Background I am trying to solve the following problem and explaining it to my students. However, my students (as well as I) think of my two assumptions contradictory. > Given 2 distinct curves $C_1: y=f(x)=e^{6x}$ and $C_2: y=g(x)=ax^2$ where $a>0$. The objective is to find the range of $a...

Can we possibly compute the following integral in terms of known constants? $$\displaystyle \int_{0}^{1}\int_{0}^{1}\frac{dxdy}{1-xy(1-x)(1-y)}$$ Some progress was already done here http://integralsandseries.prophpbb.com/topic279.html but still we have a hypergeometric function. What's your thou...

I am trying to proof $ab = gcd(a,b)lcm(a,b)$. The definition of lcm(a,b) is as follows: t is the lowest common multiple of a and b if it satisfies the following: i)a | t and b | t ii)If a | c and b | c, then t | c. Similiarly for the gcd(a,b). Here is my proof: Case I: gcd(a,b) $\neq$ 1 ...

In my recent post I have a problem with the following function: $x^4-4x^2+16$, and what I need is to find the complex roots. Here is my answer: First step, I make the substitution $x^2=y$ which involving $y^2-4y^2+16$, with $x^2=2\pm i\sqrt{12}$. Therfore: $x^4-4x^2+16=(x^2-2-i\sqrt{12})(...

Often, we find different proofs for certain theorems that, on the surface, seem to be very different but actually use the same fundamental ideas. For example, the topological proof of the infinitude of primes is quite different than the standard one, but the arguments are essentially the same, ju...