All Questions
Tagged with geometric-progressions elementary-number-theory
8 questions
3
votes
1
answer
159
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"Magic" numbers are those divided by all partial digit sums: prove that there is no infinite set of "magic" powers among the natural powers of $\ell$
For a natural number $n$, let $P_n$ the set of sums of each subset of digits in decimal notation of $n$. A number is magic if for each $s \in P_n$, we have $s \ | \ n$. Let's consider a number $\ell$, ...
0
votes
1
answer
125
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How do i verify that $272727...2727$ ($100$ digits) can or cannot be written as a perfect square?? [duplicate]
I've been stuck on this question.
I tried writing the number as as geometric progression plus $$2((10^{100}-1)/9)+5+5.10^2+5.10^4...$$
Got stuck in there.
6
votes
2
answers
324
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Showing that numbers of the form 10101010...1 are composite
I want to prove that all numbers of the form 1010101010...1 are composite except for 101.
I'm able to prove it for all numbers with an even number of ones, but I can't figure out any ideas for the ...
20
votes
5
answers
1k
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Cover $\{1,2,...,100\}$ with minimum number of geometric progressions?
In another question, posted here by jordan, we are asked whether it is possible to cover the numbers $\{1,2,\ldots,100\}$ with $20$ geometric sequences of real numbers. Naturally, we would like to ...
- 4,809
7
votes
2
answers
356
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What is known about the minimal number $f(n)$ of geometric progressions needed to cover $\{1,2,\ldots,n\}$, as a function of $n$?
What is known about the minimal number $f(n)$ of geometric progressions needed to cover $\{1,2,\ldots,n\}$, as a function of $n$?
So a geometric progression can contain at most two primes. This ...
- 26.4k
23
votes
3
answers
753
views
Is it possible to cover $\{1,2,...,100\}$ with $20$ geometric progressions?
Recall that a sequence $A=(a_n)_{n\ge 1}$ of real numbers is said to be a geometric progression whenever $\dfrac{a_{n+1}}{a_n}$ is constant for each $n\ge 1$. Then, replacing $20$ with $12$, the ...
- 15.5k
1
vote
4
answers
546
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Problems on Pythagorean triangle - sides in arithmetic (geometric) progression
Show that there is one (no) Pythagorean triangle whose sides are in arithmetic (geometric) progression.
The problem has two parts. There is one Pythagorean triangle whose sides are in arithmetic ...
- 173
16
votes
4
answers
17k
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Induction Proof that $x^n-y^n=(x-y)(x^{n-1}+x^{n-2}y+\ldots+xy^{n-2}+y^{n-1})$
This question is from [Number Theory George E. Andrews 1-1 #3].
Prove that $$x^n-y^n=(x-y)(x^{n-1}+x^{n-2}y+\ldots+xy^{n-2}+y^{n-1}).$$
This problem is driving me crazy.
$$x^n-y^n = (x-y)(x^{n-1}+...
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