An example is
\binom{n+m}{k}=\displaystyle\sum_{i=0}^{k}\binom{n}{i}\binom{m}{k-i}. It is asked to prove it for all naturals n,m,k.
They say it is enough to prove it for example just for m. The reason, which was brought forward, is that in the induction step we assume the truth of the...
Hello!
Question:
if it is asked to prove a statement A(n_1,...,n_k) for all natural numbers n_1,...,n_k, is it actually enough to check its truth by induction on just one of the counters, say n_1?
Hi! I'd like to ask the following question.
Does it make sense to take unions and intersections over an empty set?
For instance I came across a definition of a topological space which uses just two axioms:
A topology on a set X is a subset T of the power set of X, which satisfies:
1...
Ok the combination is clear:
-f(b-x)+a=f(b+x)-a
Thank you! :-)
Oh sorry I've realised that everything is ok at the very moment you've sent your post...and deleted the thing!
Hi! Brief question:
I wonder which conditions should a polynomial function of odd degree fulfill in order to be symmetric to some point in the plane.
Are there such conditions?
How can you be sure that those theorems you leave out as somewhat irrelevant will not provide your area of interest with unexpected insights and applications in the future?
Nowadays number theory(!) finds unexpected applications in physics. Now, number theory is traditionally recognized as...
Hi! I've got a question.
There is a nice formula for finding square roots of arbitrary complex numbers z=a+bi:
\frac{1}{\sqrt{2}}(\epsilon\sqrt{|z|+a}+i\sqrt{|z|-a}) where
epsilon:=sing(b) if b≠0 or epsilon:=1 if b=0.
I've just looked it up and it's nice to use it to find complex roots of...