mersenneforum.org Phi mod 11 is congruent 4 and 8
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2022-06-24, 21:22   #12
paulunderwood

Sep 2002
Database er0rr

102348 Posts

Quote:
 Originally Posted by x13420x Yes thank you 11^0/0! + 11^1/1! + 11^2/2!....... I believe this is transendental but now that I think about it I think it will go to infinity......I am doubting that this converges on a number....sorry seems I am babbling now....
I think it does converge. The terms approach 0 as n goes to infinity. I can quite make subsets of terms be bounded and making the subset sum less then some bound which clearly is by experimentation with PARI/GP. It is less than 59875. We need an analyst to step in.

2022-06-24, 21:40   #13
charybdis

Apr 2020

11001001012 Posts

Quote:
 Originally Posted by paulunderwood I think it does converge. The terms approach 0 as n goes to infinity. I can quite make subsets of terms be bounded and making the subset sum less then some bound which clearly is by experimentation with PARI/GP. It is less than 59875. We need an analyst to step in.
This is the definition of e^11...

2022-06-24, 21:46   #14
paulunderwood

Sep 2002
Database er0rr

10000100111002 Posts

Quote:
 Originally Posted by charybdis This is the definition of e^11...
I have forgotten how to prove e is transcendental. Intuitively, it cannot be written algabraically. Without me referring to my Spivak book, can you explain how it is proven transcendental? e^11 is not infinite.

Last fiddled with by paulunderwood on 2022-06-24 at 21:48

2022-06-24, 21:58   #15
retina
Undefined

"The unspeakable one"
Jun 2006
My evil lair

7·937 Posts

Quote:
 Originally Posted by paulunderwood I have forgotten how to prove e is transcendental.

https://en.wikipedia.org/wiki/Transc...transcendental

2022-06-24, 22:28   #16
paulunderwood

Sep 2002
Database er0rr

425210 Posts

Quote:
 Originally Posted by retina
For my next tricks, I have forgotten how to prove pi is transcendental, the proof of Prime Number Theorem and have never quite figured out the proof of Riemann's Hypothesis.

Last fiddled with by paulunderwood on 2022-06-24 at 22:31

2022-06-24, 22:36   #17
charybdis

Apr 2020

5·7·23 Posts

Quote:
 Originally Posted by paulunderwood For my next tricks, I have forgotten how to prove pi is transcendental, the proof of Prime Number Theorem and have never quite figured out the proof of Riemann's Hypothesis.

You might have got this confused with the proof that e is irrational, which is much simpler - I seem to recall being set it as an exercise as a first-year undergraduate.

2022-06-25, 04:48   #18
slandrum

Jan 2021
California

23×5×11 Posts

Quote:
 Originally Posted by x13420x Did you guys notice when you add the 2 results you get the modulo number plus 1? like 8 + 4 = 11 + 1
Yes, that is expected.

sqr(n) has positive and negative roots, so sqr(n) mod x will be have roots y and x-y. (sqr(n) + 1)/2 + (-sqr(n) + 1)/2 = 1

 2022-06-25, 10:03 #19 x13420x   Jun 2022 1010 Posts yes e^11 = 59874.14171519782 https://onecompiler.com/python/3y84dfvhf
2022-06-25, 11:21   #20
retina
Undefined

"The unspeakable one"
Jun 2006
My evil lair

7×937 Posts

Quote:
 Originally Posted by x13420x Would the square root of 2 be an imaginary number?
√2 ≡ 3i ≡ 8i (mod 11)

2022-06-25, 20:24   #21
oreotheory

"NT"
May 2022
U.S.

1316 Posts

Quote:
 Originally Posted by x13420x Did you guys notice when you add the 2 results you get the modulo number plus 1? like 8 + 4 = 11 + 1
Yeah isn’t that called a multiplicative inverse? (Or am I confusing things) the number x such that ax ≡ 1 (mod n)

2022-06-25, 20:31   #22
x13420x

Jun 2022

10102 Posts

Quote:
 Originally Posted by oreotheory Yeah isn’t that called a multiplicative inverse? (Or am I confusing things) the number x such that ax ≡ 1 (mod n)

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