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2
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0answers
43 views

Efficient way to count number of arithmetic progression on coloring of $\mathbb{N}$.

Consider a coloring of $\mathbb{N}$ with two colors. How many monochromatic arithmetic progressions of a fixed length $m$ (i.e. numbers of the form $a+nd$ are colored the same) are there in the subset ...
1
vote
1answer
42 views

Parity of Partition Function

Let $T(n)$ denote the number of partitions of $n$ into parts not congruent to $3$ mod $6$. Deduce that $T(n)$ is also the number of partitions of $n $ in which odd parts appear at most twice (even ...
10
votes
2answers
129 views

What are Green's almost primes?

In a general-audience talk, Ben Green explains his famous proof with Terence Tao as an application of Szemerédi's theorem, but placing the primes within a smaller set of almost-primes in which they ...
13
votes
1answer
369 views

Narcissistic numbers in other bases

It is well known that $153$ is a narcissistic number; that is, it is equal to the sum of the cubes of its digits since $153=1^3+5^3+3^3$. Other bases have similar numbers. For example, in base $3$, ...
2
votes
1answer
80 views

Numbers not of the form $x^2+My^2$

Why are there only 436 numbers not of the form $x^2+My^2$ for $x>0$, $y>1$ and $M>0$? This is A074885 from OEIS. The last number is 1875902. Can the following argument be fixed up? I ...
0
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1answer
55 views

Combinations problem with 2 contradicted answers!

If I have the digits from $0$ up till $9$: $0,1,2,3,\cdots,9$. How many 3-digit number can be made from these set of digits if the number is greater than $600$? My solution was as follows:There are ...
2
votes
3answers
398 views

Multiset Combination in Combinatorics

I want to buy a $k$-combination of doughnuts, where $k$ is any amount less than or equal to the total doughnuts available. At the bakery there are $n$ different types of doughnuts but there are ...
8
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1answer
249 views

Expectation of a Random Subset of the Roots of Unity.

Let $p$ be a prime. If $1_A(x)$ denotes the indicator function of the set $A\subset\mathbb{Z}/p\mathbb{Z}$ and $$\hat{1}_A(t):=\frac{1}{p}\sum_{n=1}^p 1_A(n)e^{2\pi i \frac{nt}{p}}$$ denotes the ...