Number Theory - an introduction - NYU Couranttschinke/teaching/Fall05/intro.pdf · 2-adic numbers...
Transcript of Number Theory - an introduction - NYU Couranttschinke/teaching/Fall05/intro.pdf · 2-adic numbers...
Numbers
Integers: Z: ...,−2,−1, 0, 1, 2...Primes: 2, 3, 5, 7, 11, ...Rationals: Q: a/b with a, b ∈ Z and b 6= 0Irrationals: 3
√2, e, π...
Number Theory - an introduction
Equations
3x + 5 = 0x2 − Dy2 = 1x2 + y2 = z2
3x3 + 4y3 = 5z3
x3 + 4y3 = 25z3 + 10t3
x4 + 2y4 = z4 + 4t4
Number Theory - an introduction
Questions
I Existence of solutions in Z or Q?
I Qualitative description of the set of solutions: finite, dense?
I Quantitative description: how many solutions?
Number Theory - an introduction
Diophantus of Alexandria
Solutions in Z of x2 + y2 = z2 are given by
x = 2mny = m2 − n2
z = m2 + n2,
with m, n ∈ Z.
Number Theory - an introduction
Pell’s equation: x2 − Dy 2 = 1, for D ∈ Z, D > 0
D = 61 : x = 1766319049 y = 226153980
D = 63 : x = 8 y = 1
D = 73 : x = 2281249 y = 267000
D = 97 : x = 62809633 y = 6377352
D = 99 : x = 10 y = 1
Number Theory - an introduction
Cubic equations
y2 = x3 + Ax + B, A,B ∈ Z
If (x1, y1), (x2, y2) are solutions, with x1 6= x2, then also (x3, y3)where
x3 = −x1 − x2 +(
y1−y2x1−x2
)2
y3 = y1−y2x1−x2
(x1 − x3)− y1.
In particular, if x1, x2, y1, y2 ∈ Q, then also x3, y3.
Number Theory - an introduction
More equations
I Euler (1769):x4 + y4 + z4 = t4 has no nontrivial solutions.(Elkies, 1998):26824404 + 153656394 + 187967604 = 206156734
I Swinnerton-Dyer (2001):x4 + 2y4 = z4 + 4t4 has no nontrivial solutions.(Elsenhans/Jahnel, 2004):4848014 + 2 · 12031204 = 11694074 + 4 · 11575204
Number Theory - an introduction
Hilbert’s problems, Paris 1900
10.“Given a diophantine equation with any number of unknownquantities and with rational integral numerical coefficients: Todevise a process according to which it can be determined by afinite number of operations whether the equation is solvable inrational integers.”
Number Theory - an introduction
Matiyasevich’s theorem (1970)/M. Davis, H. Putnam, J.Robinson
“The solubility of diophantine equations is not decidedable.”
“There is a single equation
F (t, x1, . . . , xn) = 0
with coefficients in Z, which is equivalent to all of (formalmathematics): the statement #t is provable if and only if theabove equation is solvable in x1, . . . , xn ∈ Z.”
Number Theory - an introduction
Dimension 1
axn + byn = czn
with a, b, c ∈ Z, abc 6= 0, and n ≥ 2.
I n = 2 - no solutions or infinitely many solutions
I n = 3 - none, finitely many or infinitely many solutions
I n ≥ 4 - at most finitely many solutions
Number Theory - an introduction
Diophantine geometry: curves
most important geometric invariant: genus g
I g = 0: Gauss, Minkowski, Hurwitz, Hilbert
I g = 1: Poincare, Mordell, Weil, Birch/Swinnerton-Dyer, Wiles
I g ≥ 2: Klein, ..., Faltings
Number Theory - an introduction
Cubic equations - geometry
PQ
P ⊕Q
This is how one “adds” rational points.
Number Theory - an introduction
Dimension 2
axn + byn = czn + dtn
with a, b, c , d ∈ Z, abcd 6= 0, and n ≥ 2.
I n = 2 - no solutions or a dense set of solutions
I n = 3 - no solutions or a dense set of solutions
I n ≥ 4 - ???
Number Theory - an introduction
“Imagine this surface is a building. Isn’t one of the lines a goodplace for a balcony and the other one a great place to i nstall an
escalator straight from the bottom to the top?”
Number Theory - an introduction