r/numbertheory

Came up with this one for fun, no idea if it's been posted before somewhere. Fair warning, I'm not amazing at math, just got curious about this one and worked through it slowly. Mostly wanted to share because I liked how a silly real-world setup ended up landing right on top of φ(n).

The setup

In an online poll, viewers vote either "Yes" or "No," and the result is displayed only as a percentage rounded to exactly two decimal places (e.g., 41.27%). The total number of votes is not shown. Assume that for any percentage displayed, the actual vote tally is the minimum possible whole number of votes that could have produced that exact percentage.

The question

Out of all possible displayed percentages (from 00.01% to 99.99% in steps of 0.01%), how many of them require the full 10,000 voters as a minimum? And which displayed percentages are those, intuitively?

>!Where coprimality comes in!<

>!A displayed percentage X.XX% corresponds to the fraction XXXX/10000. The minimum number of voters needed to produce that exact ratio is 10000 / gcd(XXXX, 10000). So the minimum hits its maximum (10,000) exactly when gcd(XXXX, 10000) = 1, i.e., when the numerator is coprime to 10,000.!<

>!Two numbers are coprime when they share no prime factors. Since 10,000 = 2⁴ × 5⁴, its only prime factors are 2 and 5. So XXXX is coprime to 10,000 if and only if XXXX is odd AND not divisible by 5. That's a clean shortcut, you don't have to actually factor the numerator at all, you just check the last digit.!<

>!Where Euler's totient comes in!<

>!The count of integers from 1 to n that are coprime to n is exactly Euler's totient function φ(n). For n = 10,000:!<

>!φ(10000) = 10000 × (1 − 1/2) × (1 − 1/5) = 10000 × 0.5 × 0.8 = 4,000!<

>!So exactly 4,000 displayed percentages require the full 10,000 voters as a minimum. That's 40% of all possible X.XX displays.!<

>!The pattern generalizes nicely. If you display to d decimal places, the max minimum is 10^(d+2), and the number of splits tied at that max is φ(10^(d+2)) = 0.4 × 10^(d+2). Always exactly 40%, because the prime factorization of any power of 10 only involves 2 and 5, and (1 − 1/2)(1 − 1/5) = 0.4.!<

>!The part I thought was nice!<

>!The reason the answer is always 40% (regardless of how many decimal places you display) is that 10 only has two prime factors. If we counted in some weird base where the denominator had more prime factors, the proportion of "hardest" splits would drop. The fact that our base-10 display gives such a clean answer is a small accident of the base we count in.!<

Curious if anyone sees a slicker way to frame the general result, or if there's a related problem I should look at. Also happy to be told this is a well-known exercise and I just reinvented it.

This is a repost from r/math since I don't use reddit I can't post there. I think this is the most appropriate sister thread.

So a few years ago I noticed a pattern about differences of squared numbers. However, I failed to find anything about it. It just popped into my head again, and I am not conceited enough to think I invented 'new math' or whatever. So someone tell me this is a thing and I am just ignorant.

The concept goes as follows... the difference between the additive amounts of squared numbers is always two more than the last. At least when moving up integers. When moving down it decreases by 2. This is at the exclusion to 0^2.

Exemplified as follows:

1^2 | 2^2 | 3^2 | 4^2 | 5^2 |
1 4 9 16 25
+3 +5 +7 +9
+2 +2 +2

If what I put above is readable see how the difference of 3^2 (9) and 4^2 (16) is 7, then the difference of 4^2 (16) and 5^2 (25) is 9. Then notice how the difference of 7 and 9 is 2. And how it is always 2 between adjacent sets of squared results. This pattern goes on for as far as I checked.

Title: Conjecture on repeated multiplication and eventual convergence to integers

​I've been investigating a pattern regarding repeated multiplication and decimal convergence. After testing over 1,000 cases, I haven't found a counter-example yet and I’m looking for a mathematical explanation or a case where this fails.

​The Process:

Take two decimal numbers, a and b. Multiply a by b, then take the result and multiply it by b again. This continues in a feedback loop (a_{n+1} = a_n \times b).

​The Rules:

To avoid trivial cases or infinite loops, I’ve set these constraints:

​The first and last digits of both numbers cannot be 1.

​The first and last digits of a must be different from the first and last digits of b to avoid symmetry.

​The Observation:

No matter how many digits there are after the decimal point, the results eventually "explode" in scale and, in every test I've run, eventually become a perfect integer (whole number). Even with complex decimals like a = 0.455433015 and b = 5.314541154, the decimal tail eventually clears.

​The Question:

Is there a known theorem that explains why these types of sequences would consistently hit an integer? Since the numbers grow exponentially, it becomes difficult to track with standard floating-point precision on a home setup. I’m interested to know if this has a specific name in number theory or if someone can identify a specific pair that would refuse to converge to a whole number.

The supersignum unit g is defined as a bridge between hyperbolas and circles, its chaotic set or unit that i made, it starts with i, a concept everyone knows, then i²=-1, then we suddenly get j, a hyperbolic number where j²=1, but g²=±1, lets see their powers

i²=-1

So

i³=-i

This may look weird but its part of the plan

i⁴=1

Its a full rotation!

Now j

j²=1

1×j=j³=j

That was a fast loop

Now g

g=g

g²=±1

g³=±g (logically)

But whats g⁴?

±1×±1=1 so g⁴=1

And ±g×g=1 may look weird, but its normal, ±1×g×g,±1×±1, see! We get the same result

So lets find what set is g

g²={-1,1}

we take the square root and assume √1=j since j²=1 as an soloution

√g² take root

√g²={√-1,√1}

g={i,j}

Wow!

Extra : if we encounter an i during the i path and j says the same, for example (iπ)/2 and (jπ)/2, we can say (gπ)/2 in ln(g) because it happens

Dicoilic numbers: this is where the fun begins, its not supersignum numbers, but it has 3 dimensions

A dicoilic number is a number a+bw+cs

|a+bw+cs|=√|a²+b²c|

You can do stuff with dicoilic numbers

Dicoilic numbers are a+bw+cs

W and s are not regular units and 0s≠0 to prevent epsilon=w

Lets start with a few stuff

i=w+s

j=w-s

epsilon=w±0s (+ and - are interchangeable)

Lets find the hypercomplex unit k

We know k=ij

That means (w+s)(w-s)

That means k=w²-s²

We cant exactly find w² and s² but it does have some algebra

i²=w²+s²+2sw

i²=(w+s)²=w²+s²+2sw

i=w+s

j=w-s

This system is communitave

That means the hypercomplex unit k

k=ij

k=(w+s)(w-s)

k=w²-s²

j²=w²+s²-2sw

This means j²+i²=0

And w²+s²=0

Whaaat

w²=-(s²)

Amd i²-j²=-2 or 4ws

That means 4ws=-2 divide

2sw=-1

Lets check if this is consistent

I²=s²+w²+2sw

S²+w² is 0

i²=2sw

2sw=-1

CONSISTENT!

And check for j

j²=1

j²=w²+s²-2sw

j²=0-2sw

j²=0-(-1)

j²=1

LOL

In dicoil numbers, there is a concept called dicoilic form, every hypercomplex number and imaginary can be expressed in a dicoilic form

i=w+s

j=w-s

k=w²-s²

Epsilon=w+0s

If we want to take the dicoilic form of , say, 1+i, we put the real part down first

1

Then we take the number

i=w+s

Then we get 1+w+s