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2d
revised Find the number of flags of different types using induction
added 509 characters in body
2d
answered Find the number of flags of different types using induction
2d
comment Is there a name for the closed form of $\sum_{n=0}^{\infty} \frac{1}{1+ a^n}$?
Maple, Mathematica (and others) will give exotic "closed forms" that are essentially tautologous.
2d
comment The existential theory is undecidable
I should also mention Alexandra Shlapentokh (my spelling may be off).
2d
comment The existential theory is undecidable
I imagine you are gathering a bibiliography on the subject. An active person has been Bjorn Poonen.
2d
comment The existential theory is undecidable
By the way, the big open problem in this game, that people have been working on for more than 40 years, is whether there is an algorithm for determining solvability of Diophantine equations in rationals.
2d
comment The existential theory is undecidable
I will (perhaps tomorrow, busy time, long term visitors), deal with $\psi$, it is the subject of another of your questions. The stuff about $-1$ was really not relevant to your main question, about existential definability, it was just a side comment. When we write (i) $P_1(\bar{u})=0)\land P_2(\bar{v})=0$ (with existential quantifiers in front) we have an existential formula. If the sum of squares is $0$ iff each term is $0$ (true in the rationals, the reals, but not the complexes) we can replace the $\land$ by $P_1^2+P_2^2=0$, which makes it look more Diophantine.
2d
comment What is the definition of a Critical Point?
Different people use the term in somewhat different ways.
2d
comment Use comparison test to determine convergence
A start: From the power series $\sinh x=x+x^3/6+\cdots \gt x^3/6$. Or limit comparison with the function $e^{-x/2}$.
2d
comment Generators in group $Z^*_{p}$
The "lengthy" way is to find the powers of $2$ modulo $19$. You will get $18$ different ones. There is a useful and not hard theorem that if $p$ is a prime, and $a^{(p-1)/q}\not\equiv 1\pmod{p}$ for every prime divisor $q$ of $p-1$, then $a$ is a generator. So all we need to do is to check that $2^9\not\equiv 1\pmod{19}$ and $2^6\not\equiv 1\pmod{19}$.
2d
comment Modular Quadratic Equation
Has (Hensel) lifting been done in your course already?
2d
revised Find a recursion (combinatorial)
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2d
revised Find a recursion (combinatorial)
added 477 characters in body
2d
comment Elementary number theory, compute this sum
@wythagoras: Thank you, you answered OP's question in a notationally much clearer way than I did.
2d
comment Elementary number theory, compute this sum
Let $N=p_1^{a_1}\cdots p_k^{a_k}q_1^{b_1}\cdots q_l^{b_l}$ where the $p_i$ are primes of the form $4t+3$ and the $q_i$ are primes of the form $4t+1$. If all the $a_i$ are even, then $N\equiv 1\pmod{4}$.
2d
answered Elementary number theory, compute this sum
2d
comment Show that $\int_{\pi}^{\infty} \frac{1}{x^2 (\sin^2 x)^{1/3}} dx$ is finite.
The integral from $n\pi$ to $(n+1)\pi$ of $(\sin^2 x)^{-1/3}$ seems to be controllable.
2d
answered Find a recursion (combinatorial)
2d
comment How to prove the limit exists for function of two variables?
The first reasoning is correct, well done. The second (Squeeze) has an easily fixed flaw at points $(0,y)$.
Jul
28
comment Elementary Substitution in Solving Equations - Why it works
When we square, or take the sine, or make some other not one to one transformation, verification that we have not introduced an extraneous root becomes necessary.