🧵 The Mario Conjecture
Anonymous at Thu, 18 Jul 2024 11:40:20 UTC No. 16286994
>If you convert the numbers in Pi to binary, somewhere in the endless string of 1's and 0's you will find the binary of Super Mario Bros
is this true?
🗑️ visit soygem.party at Thu, 18 Jul 2024 11:46:55 UTC No. 16287012
>>16286994
That's an open problem. I believe that if pi is proved to be a normal number this is true, but no one proved pi is normal yet. Proving numbers are normal is very hard.
https://en.wikipedia.org/wiki/Norma
Anonymous at Thu, 18 Jul 2024 12:07:30 UTC No. 16287079
Depends if it's a normal number or not.
>>>/sci/sqt/
Anonymous at Thu, 18 Jul 2024 14:33:31 UTC No. 16287248
>>16286994
Pi's normality is unknown. Being irrational doesn't require every sequence to appear. For example, consider this number where the number of zeroes between ones keeps increasing: 0.101001000100001000001... That number is irrational, but you'll never find a 2 in it.
Anonymous at Thu, 18 Jul 2024 14:34:15 UTC No. 16287251
>>16286994
>is this true?
No probably not but likely cannot be proven with 100% certainty since pi doesn't have a known pattern (as far as I know)
Anonymous at Thu, 18 Jul 2024 18:43:24 UTC No. 16287544
>>16286994
by the way, pi has infinite decimals in base 10, but how do we know the amount of decimals is infinite in base 2 or in base 16? even if they are infinite, maybe they are periodical, as in the numbers aren't random anymore and they repeat themselves
Anonymous at Thu, 18 Jul 2024 18:45:31 UTC No. 16287547
>>16287251
What about another number that can be multiplied by pi to create the proper sequence. How would that second number be found?
Anonymous at Thu, 18 Jul 2024 18:57:54 UTC No. 16287567
>>16287079
If it's normal, then it the claim follows, yes. But if it isn't normal, it's not ruled out.
Anonymous at Thu, 18 Jul 2024 19:01:33 UTC No. 16287576
>>16286994
Absolutely false.
The binary of any computer program contains many repeats that could not be generated by the binary representation of an irrational number that never repeats.
Anonymous at Thu, 18 Jul 2024 19:02:57 UTC No. 16287578
>>16287576
but I looked at pi and saw the pattern 69 come up a bunch, how's that possible if it doesn't repeat?
Anonymous at Thu, 18 Jul 2024 19:08:33 UTC No. 16287586
>>16287544
If a number does not have infinite decimals in all bases, wouldn't that make it a rational number?
Anonymous at Thu, 18 Jul 2024 19:11:59 UTC No. 16287595
>>16287544
>but how do we know the amount of decimals is infinite in base 2 or in base 16?
If x is finite in base b, then there exists a_0 integer, a_1, ... a_n natural numbers smaller than b such that x = a_0 + a_1/b + a_2/b^2 + ... + a_n/b^n = (a_0 b^n + a_1b^(n-1)+.... + a_n)/b^n which is rational.
Pi is know to not be rational. Therefore in any integral base it has an infinite expansion.
Anonymous at Thu, 18 Jul 2024 19:15:14 UTC No. 16287603
>>16286994
It definitely is true if you just consider a large enough amount of digits. For example, if you have the first TREE(TREE(Graham's number^(TREE(3)))) digits of pi in base-2, it's almost 100% certain that there weill be a pirated version of Windows 10 somewhere in there, probably more than a billion times with that many digits.
Anonymous at Thu, 18 Jul 2024 19:20:15 UTC No. 16287610
>>16287544
Any rational number that can be expressed using a finite number of digits in base 2 can be expressed using the same number of digits in base 10 because 2 is a factor of 10.
Any rational number that can be expressed using a finite number of digits in base 16 can be expressed using 4 times the number of digits in base 10 because 16 is a factor of 10^4.
You would have a point for bases such as three, because any rational number, expressible using a finite number of digits in base 3 and in the range ]0, 1[, is necessarily only expressible in base 10 using an infinite number of digits. BUT, the length of the repetend will be equal to the number of digits used in base 3.
>>16287547
I recall hearing that pi^^4 might be an integer