All Questions
10
questions
2
votes
3
answers
177
views
Is $1 \times 1 + 2 \times 2 + 3 \times 4 + 4 \times 8 = 49$ a coincidence? (Is $\sum_{i=0}^k(i+1)2^i$ ever a square again?)
When watching a gaming video, I noticed an intriguing fact:
$$
1 \times 1 + 2 \times 2 + 3 \times 4 + 4 \times 8 = 49,
$$
which is a square number.
I asked myself, is this a coincidence or not? ...
- 4,051
0
votes
1
answer
105
views
Evaluation sum and its asymptote $\sum_{s=1}^{N} \sum_{t=1}^{N} \left[{\sqrt{4t-s^2} \in \mathbb{Z}}\right]$
I am working on the evaluation of $$S \left({N}\right) = \sum_{s=1}^{N} \sum_{t=1}^{N} \left[{\sqrt{4t-s^2} \in \mathbb{Z}}\right]$$ and its asymptotic expansion where $N \ge 1$. Here $\sqrt{4t-s^2} \...
- 967
6
votes
1
answer
220
views
Only $1$ and $4900$ are squares as $1+4+9+\ldots+ n^2$
I encountered this fact yesterday: $1$ and $4900$ are the only squares as the sum of $1+4+9+\ldots +n^2$. I was trying to solve this problem using my knowledge of elementary number theory. I reduce it ...
- 111
3
votes
0
answers
318
views
Which natural numbers are the sum of three positive perfect squares?
In this question : Which positive integers $n$ can be expressed as $n=a^2+b^2+c^2$ , but not with positive $a,b,c\ $?
I asked for the classification of the natural numbers being the sum of three ...
- 85.1k
4
votes
0
answers
117
views
Is $p^2+q^2+r^2=3^k$ with primes $p,q,r$ solvable for every odd positive integer $k\ge 3\ $?
For the positive odd integers $3\le k\le 25$, the equation $$p^2+q^2+r^2=3^k$$ with primes $p,q,r$ is solvable. Here is one solution for every exponent , calculated with PARI/GP :
...
- 85.1k
3
votes
0
answers
162
views
Which positive integers $n$ can be expressed as $n=a^2+b^2+c^2$ , but not with positive $a,b,c\ $?
Let $S$ be the set of positive integers $n$, such that the equation $$a^2+b^2+c^2=n$$ has a solution in non-negative integers but not in positive integers. Since $n\in S$ if and only if $4n\in S$, we ...
- 85.1k
7
votes
2
answers
794
views
Classification of the positive integers not being the sum of four non-zero squares
It is well known that every positive integer is the sum of at most four perfect squares (including $1$).
But which positive integers are not the sum of four non-zero perfect squares ($1$ is still ...
- 85.1k
-1
votes
1
answer
170
views
2041 distinct natural numbers such that the sum of their squares is a perfect square
Determine if there are 2041 distinct natural numbers such that the sum of
their squares is a perfect square.
Can anyone please help me to solve this problem?
1
vote
4
answers
322
views
How do I solve Problem 1 of the International Olympiad of Metropolis?
The question is: Find all $n$ so that there exists $n$ consecutive numbers whose sum is a square!
My method to solve the problem: I would try to look at the values modulo $n$, and there I see there ...
user368587
36
votes
9
answers
4k
views
Is difference of two consecutive sums of consecutive integers (of the same length) always square?
I am an amateur who has been pondering the following question. If there is a name for this or more information about anyone who has postulated this before, I would be interested about reading up on it....
- 379