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Finite integral with goniometric functions, $int_0^{infty} frac{8sin^4(pi f...
I have difficulties trying to find an algebraic solutions of the following integral: The $tau$ in this formula is an integer, which is a very important fact because only then this integral is...
View Articlehow to tell a fraction in denominator or numerator should be substituted with...
Suppose we have equations as follows (A, C and B are all integers and $gcd$=greatest common divisor). $$R_1 = frac{Atimes C}{B} hspace{2cm} R_2 = frac{Atimesfrac{C}{gcd(B,C)}}{frac{B}{gcd(B,C)}}$$ Now...
View ArticleA sequence of subsets of $Bbb Z$ not containing nontrivial subgroups
Is there a sequence $(A_n)$ of subsets of $Bbb Z$ such that always ${a-bmid a,bin A_{n+1}}$ is a proper subset of $A_n$ and no $A_n$ contains an infinite subgroup of $(Bbb Z,+)$?
View ArticleIs this version of Lagrange's four-square theorem true?
Lagrange’s four-square theorem states that any natural number $n$ can be represented as the sum of four integer squares.i.e. $n = a_1times a_1 + a_2times a_2 + a_3times a_3 + a_4times a_4$ Question: Is...
View ArticleA positive integer $n$ is such that...
A positive integer $n$ is such that $$1-2x+3x^2-4x^3+5x^4-…-2014x^{2013}+nx^{2014}$$ has at least one integer solution. Find $n$.
View ArticleValue of an expression with an impossible predicate
let $a = (a_1, a_2, a_3, ldots , a_n)^T$ for some arbitrary large (irrelevant to the question) value of $n$. and let $i,k in mathbb{Z}$ What would the value of the following expression be when $i = 0$?...
View Articleg.c.d of two numbers [on hold]
Let $n$ and $m$ be two integer numbers, is one of the following always true? 1) $gcd(n,m)=1$ 2) $gcd(n−1,m)=1$ 3) $gcd(n,m−1)=1$ 4) $gcd(n−1,m−1)=1$
View ArticleInduction Proof from Thomas Judson book on abstract algebra
I’m trying to prove $$^nsqrt{a_1times a_2times…times a_n}leq frac{1}{n}sum_{k=1}^na_k, quad a_iin mathbb{Z}^+$$ by Induction. The case is true for $n=1$ so I assumed true for $n=k$. I then tried...
View ArticleSimplify P(n), where n is a positive integer : $ P(x)=sum limits_{k=1}^infty...
This is what I have tried, but I don’t know what to do next, so I need help : $ P(x)=sum limits_{k=1}^infty arctanleft(frac{x-1}{(n+x+1)sqrt{n+1}+(n+2)sqrt{n+x}}right). $ $...
View ArticleWhat is the best time complexity of checking the inequality $a_1x_1 + cdots +...
We know that all the coefficients $a_1, a_2, ldots , a_m$ are integer. Also, $K$ is an integer number. I only need to know if the inequality has a integer solution or not. It means that there is no...
View ArticleCoin Change Problem with Fixed Coins
Problem: Given $n$ coin denominations, with $c_1<c_2<c_3<cdots<c_{n}$ being positive integer numbers, and a number $X$, we want to know whether the number $X$ can be changed by $N$ coins....
View ArticleWhat phenomenon is this? $(2Bbb{Z} + 1)cup 3Bbb{Z} = 2Bbb{Z} cup 3Bbb{Z} + 3$
$(2Bbb{Z} + 1)cup 3Bbb{Z} = 2Bbb{Z} cup 3Bbb{Z} + 3$ Proof: $$ begin{align*} 2Bbb{Z} &= bullet circ bullet circ bullet circ bullet circ dots \ 3Bbb{Z} &= bullet circ circ bullet circ circ...
View ArticleNature of primes as the building blocks of integers?
It is considered standard in mathematics that all integers can be expressed as the product of primes: $$n=p_1^{a_1}p_2^{a_2}…p_k^{a_k}$$ Where $p_i$ is prime and $p_{i-1}<p_i$ and $a_i$ is and...
View ArticleIMO 2007 problem 1 [on hold]
Real numbers $a_1, a_2, cdots, a_n$ are given. For each $i$ ($1 leqslant i leqslant n$) define $$ d_i = max {a_j : 1 leqslant j leqslant i } – min {a_j : i leqslant j leqslant n } $$ and let $$ d = max...
View ArticleDoes the following equation have a positive integer solutions for $p$?
Q. Does the following equation have a positive integer solutions for $p$? If are unable to obtain an answer for $p$, explain why this is the case. If you are able to find one case where the equation...
View ArticleFunctions Mapping Integers to Zero?
I am looking for functions such that: $z∈$ Z ⇔ $f(z)=0$ That is to say, functions that map from Z to the zero set. One example is $f(z)=sin(πz)$. EDIT: To narrow the possible group of functions, the...
View ArticleLet $a,b in mathbb{Z}$. Prove that if $b = qa + r, q,r in mathbb{Z}$, then...
This is a lemma from Rotman’s book “Advanced Modern Algebra” Let $a$ and $b$ be integers(and so are $q$ and $r$). I need to prove that if $b = qa + r$, then $gcd(a,b) = gcd(r,a)$. Not sure how to...
View ArticleAlternative decimal number representation
In class we had an overview of different representations of numbers. Some examples were: decimal, roman numerals, binary, Redundant binary representation, church numerals… Then the teacher gave an...
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