24
votes
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21
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Can you find a 3x3 white square somewhere in this relatively prime graph?
One solution I've found
Construction
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Self-referential sequence that is sometimes powers of two
The sequence is defined as:
For example,
This explains the note that we the sequence can't start with two of the same number because
To answer the questions:
What comes after $115$ in the second ...
- 26.3k
15
votes
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An Almost-squarish set of numbers
The first few almost-square numbers are: $3,8,15$
And we notice that $3\times 8=24$ is the next almost-square number.
As such, we have the almost-squarish set $\{1,3,8\}$ with 3 distinct elements.
If ...
- 13.4k
14
votes
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13
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Vector Sum of Pythagorean Triples
The graph of real-valued Pythagorean triples $x^2+y^2=z^2$ forms an infinite cone if we restrict to $z>0$:
A sum of $n$ independent vectors on this cone is $n$ times their average, which lies ...
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12
votes
How abundant can a number get?
Does such a number exist?
We require a few priors:
Where to begin?
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Numbers whose product of digits is a multiple of sum of digits
Three numbers
Observe that:
Moreover:
So we can start by finding...
At this point it is worth noting that:
So we search and find that
Now observe that:
So:
With this much flexibility for ...
- 759
11
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An Almost-squarish set of numbers
The existence of a "Diophantine quintuple" was an open question for many years.
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9
votes
Super Star Numbers
This answers b)
Given four distinct primes p,q,r,s and a positive integer X, the number $p^{12}q^{12}r^3s^3X$ is superstar by the following construction:
Given positive n choose 4n distinct primes $...
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Square Sum Problem Summing 3 consecutive digits along the line
An exhaustive search has found that the lowest value of n for which this is possible is:
With the sequence:
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Save now! All the digits at half the price
Let's define some terms:
Now we can begin.
Let's break the problem down now.
Now for the final count!
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Primeable numbers
Allowing for zeros to be discarded to form primes with fewer digits, the first sequence containing ten or more consecutive non-primeable numbers is:
Related sequences:
Searching all numbers with six ...
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7
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An Almost-squarish set of numbers
Some hand analysis first.
This is sufficient to prove there are infinitely many 4 element sets.
These are NOT all of the 4 element options.
I believe you can start with any number (at all), then ...
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Super Star Numbers
There are 75,863 Super Star Numbers not greater than one million. Within this range there are 237 pairs of consecutive Super Star Numbers, yet no triplets.
The first pair, with solutions shown below, ...
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5
votes
Save now! All the digits at half the price
My guess is:
First, observe that the first digit of $x$:
Then, the reasoning goes this way:
I'm not 100% sure of my calculus but this is how I did it:
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5
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An Almost-squarish set of numbers
While trying to make my own code to generate all the groups of 4 I found a pattern. I don't know if we could expand it to eventually find a 5th element but i found that if you include 1, most of the ...
5
votes
Vector Sum of Pythagorean Triples
By definition of (Euclidean) length, 2-dimensional vector $v_j:=(a_j,b_j)$ has length $|v_j|=c_j$ if and only if $a_j^2+b_j^2=c_j^2$ ($c_j\ge 0$). By repeated application of the
we have for any ...
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5
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Intermingled primes
The 6 different primes are:
Deductions (I solved it with regular expressions in a list of all prime numbers from 101-997):
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Primeable numbers
Let's try to understand the probability of very large unprimeable numbers. To do so, let's start by looking at how many permutations a number has. If a number has only a single unique digit, it has ...
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Self-numbers and repunits
Are there infinitely many repunits which are not Self-numbers?
Are there infinitely many Self-numbers which are repunits?
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4
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Dead By Daylight
This may be a partial answer, because I don't see how to reverse the "encryption", and I'm not enough of a mathematician to confidently say this transform is not reversible.
Explain how the ...
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Smallest 3x3 Magic Square of different square sums
answer 1 (square sums)
This is the smallest because...
answer 2 (triangular sums)
This is smallest because...
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4
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4
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Save now! All the digits at half the price
A bit messier than I'd like, but here goes.
As other answers have noted, the doubleable PD10 numbers we want are those where
Let's show that every such number can be specified by:
for an overall ...
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4
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What do 84, 96 and 108 have in common?
Partial answer
Update: With the hint about "first k divisors", I found out the first 9 numbers satisfy being the product of 6th and 7th (smallest) divisors, however other numbers outside the ...
- 693
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