Phi mod 11 is congruent 4 and 8

[paulunderwood]

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.

[charybdis]

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...

[paulunderwood]

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.

[retina]

Originally Posted by paulunderwood

I have forgotten how to prove e is transcendental.

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

[paulunderwood]

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.

[charybdis]

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.

[slandrum]

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

[x13420x]

yes e^11 = 59874.14171519782

https://onecompiler.com/python/3y84dfvhf

[retina]

Originally Posted by x13420x

Would the square root of 2 be an imaginary number?

√2 ≡ 3i ≡ 8i (mod 11)

[oreotheory]

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)

[x13420x]

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)

The additive inverse....