Transcendental Number Theory

Hermite proved e transcendental in 1873; Lindemann proved pi in 1882, closing squaring the circle. Baker received the 1970 Fields Medal for effective bounds on logarithmic linear forms.

• Cryptography: digits of pi and e seed pseudorandom generators
• Diophantine equations: Baker's theorem finds all integer solutions effectively
• Algebraic geometry: periods of varieties are conjectured transcendental

Предварительные знания

• Previous lesson

Liouville Numbers and the Transcendence of e

Charles Hermite in 1873 proved e is transcendental; Lindemann in 1882 proved pi, immediately closing squaring the circle -- a 2500-year open problem. Liouville in 1844 built the first explicit transcendental: L = sum_{k>=1} 10^{-k!} = 0.11000100...

Gelfond-Schneider Theorem

Hilbert in 1900 posed Problem 7: is 2^sqrt(2) transcendental? In 1934, Gelfond and Schneider independently proved: if alpha is algebraic, not 0 or 1, and beta is algebraic irrational, then alpha^beta is transcendental. Alan Baker received the 1970 Fields Medal for effective lower bounds on linear forms in logarithms.

Key Ideas

• Algebraic: root of polynomial over Q; transcendental: none
• Liouville: too-good rational approximations imply transcendence
• Hermite 1873: e transcendental; Lindemann 1882: pi transcendental
• Gelfond-Schneider 1934: alpha^beta transcendental for alg alpha!=0,1 and irrat beta

Further Directions

These ideas open paths to deeper number theory.

• arith-28-p-adic — extends

Вопросы для размышления

• Give a concrete example.
• How does this connect to other areas of number theory?