Pi hiding in prime regularities

Three mysterious Math Objects:

  • Pi, 
  • Complex Numbers, 
  • Prime Numbers

are hiding in circle.


100-digit Pi

One fine day when we reach above 80 years old, if the doctor accuses us of having dementia, then prove the doctor wrong by shocking him with 100-digit Pi memory 🙂

With Chinese single-syllable sound for numbers, better still if can sing it as a song, memorizing 100-digit pi is easy!


Transcendental Number

Transcendental numbers: e, Π, L…

What about e^{e},  \pi^{\pi} ,\pi^{e}  ?

Aleksander (Alexis) Osipovich Gelfond (1906-68):

Gelfond-Schneider Theorem

a^ b transcendental if
a is algebraic, not 0 or 1
b irrational algebraic number

\sqrt{6}^{\sqrt{5}},  3^{\sqrt{7}}
Hilbert Number: 2^{\sqrt {2}} (Hilbert Problem proven by Gelfond}

Is log 2 transcendental ?
[log = logarithm Base 10]

10^{log 2} = 2

1) Sufficient to prove log 2 irrational
Assume log 2 rational
log 2= p/q, p and q integers
10^ {log 2} = 2 = 10^ {p/q}
raise power q
2^{q} = 10^{p} = (2.5)^{p}
2^{q} = 2^{p}.5^{p}

Case 1: p>q
1 = 2^{p-q}.5^{p}
=> False

Case 2: q>p
2^{q-p}= 5^{p}
Left is even : 2^{m} \text { = even}
Right is odd: 5^{n} \text {= ....5}
=> False

Therefore p,q do not exist,
=> log 2 irrational

Reference: Top 15 Transcendental Numbers: http://sprott.physics.wisc.edu/pickover/trans.html