Théorie des graphes Arithmétique

+− × /
1641 −1994 x, y, z n ∈N∗xn+yn=zn
n= 2 (x, y, z) (3,4,5) (5,12,13)
n > 2
Z
a b a b a |b
b a
a, b c Z∗
a|b b |c⇒a|c
a|b a |c⇒a|(nb +mc)∀n, m ∈Z
a|b b |a⇒(a=b a =−b
a∈Zb∈N∗(q, r)a=bq +r0≤r < b
a b q r

185 31
23456 671
a b N∗d∈N∗
c∈N∗c a b c d
a b N∗m∈N∗
c∈N∗c a b c m
a b a ∧b) 1
15 12 15 12
15Z+ 12Z={au +bv, u ∈Z, v ∈Z}
a b
a b 2
a∧b=b∧a a ∧(b∧c) = (a∧b)∧c
a b
(q, r) (a, b) (q, r)
a b
144 ∧39
u v
d=au +bv 2
1
2

u v 3 = 144u+ 39v
d=a b u v d =au +bv a = 4235 b= 42
144 ∨39 =
a b ×a b =|ab |

ax +by =c
S ax +by =c a b c
d=a∧b a =da1b=db1a1∧b1= 1
c d
c=dk
(u, v)ax +by =d
(x0, y0)ax +by =c
S
4235x+ 42y= 15,4235x+ 42y= 14.

T(k)k
|a|a|b|b
k:= max(|a|,|b|)
|a|×|b| × 10−9s.
5×inf(|a|,|b|)
ax +by =c
100
p
p2100
23579
11 13 15 17 19
21 23 25 27 29
31 33 35 37 39
41 43 45 47 49
51 53 55 57 59
61 63 65 67 69
71 73 75 77 79
81 83 85 87 89
91 93 95 97 99
499
n
n√n