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数字的奇妙：素数

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Entries A to Z. ' J) s8 V9 t) J8 T
abc conjecture. ' i& P. n# L$ W
abundant number. 1 `+ p6 e* s% z5 I, m$ X, w8 P
AKS algorithm for primality testing. 2 D: r# k3 ?, X
aliquot sequences (sociable chains).
5 G3 e) m9 V& s2 c7 E* a% jalmost-primes.
# ~# |1 X9 X7 n+ aamicable numbers.
7 [$ a% i2 M7 n- ?- b$ B) p6 mamicable curiosities. ! d: k, r% _- z2 |3 W# M( h
Andrica’s conjecture. # z% q. a. P4 D$ s
arithmetic progressions, of primes.
+ b* N7 O+ G- N" \+ Q2 [6 ?Aurifeuillian factorization.
 v8 c' w6 b1 d* u: f, E) Uaverage prime. ; G- z$ H6 i' d$ o0 I. r7 x
Bang’s theorem.
( A% O3 e$ ^6 |Bateman’s conjecture.
- _8 Q9 B [8 q2 U0 w6 L9 c: PBeal’s conjecture, and prize. & }1 t- R( t' m( I# D. b
Benford’s law.
& `, e2 V2 E# d9 NBernoulli numbers. ! e. d. c5 y# V. d; I( B
Bernoulli number curiosities. . F% i; r$ Z3 ?2 E- m
Bertrand’s postulate. 4 J6 S2 d1 _5 C. f! X* [) N7 i0 o, J
Bonse’s inequality. , |: }+ j& B' r( _1 L9 I
Brier numbers.
Brocard’s conjecture.
1 ?2 h! l: |, }5 kBrun’s constant.
' h+ L6 B2 P( eBuss’s function.
" a: |% I1 t( N3 q# t4 R& ]! fCarmichael numbers. 8 \% ^4 n4 B* Q# y
Catalan’s conjecture. 3 u) W5 y% p9 |; ]+ x% t( T8 \
Catalan’s Mersenne conjecture.
& X8 T6 h4 |) V! L( w! W3 qChampernowne’s constant. q- R: I R' }3 E) f
champion numbers.
% \+ e0 h* w6 i9 r9 M" RChinese remainder theorem. ; ]! U, w2 @( a$ {' P1 B/ \
cicadas and prime periods.
, Z$ o+ @; Q s* B; Gcircle, prime. % x/ J1 T6 N3 Q. O# P, H
circular prime. 4 U7 a; h/ l, n& H: z
Clay prizes, the. 7 h* x1 R, k+ b5 N8 ?* \1 p
compositorial. $ [5 {4 F/ v7 l# D
concatenation of primes. + k, v* h4 h1 I- ?; |" M
conjectures. W$ ]4 c- x( T" n' e$ R/ B1 ]
consecutive integer sequence.
' G( Z$ n- B0 h+ G2 a' U% Aconsecutive numbers.
$ r/ | g, |# \" Aconsecutive primes, sums of. 1 E- r0 V" ~% X0 ] i+ s
Conway’s prime-producing machine.
- e4 A4 X; C* p4 k: pcousin primes.
5 R# K8 N9 e/ h2 V X; ^+ r/ RCullen primes.
) L E N7 H7 U" f, T1 A% c3 P0 [Cunningham project.
4 u! k- `* f3 ECunningham chains. , R8 ?+ @ v( ?% ?8 q: j+ m# H
decimals, recurring (periodic).
3 F+ ]: S M4 q: D( W5 k1 Z. bthe period of 1/13.
, Y7 V' q% a$ b# d2 H+ t/ s7 Ocyclic numbers. 5 j4 `7 T. y1 B0 t: K
Artin’s conjecture. g8 X/ O8 Q; n' C& v
the repunit connection. * a, A. X# J1 y* @+ _3 L
magic squares. . v V' t# M* M5 f w& H* l
deficient number.
8 N$ Y) C& p, ], |9 U: X6 `deletable and truncatable primes.
( O% f/ Y+ Y) p* ?8 j+ A! @Demlo numbers.
) k3 @/ r- E, k" o. i$ Tdescriptive primes. 3 ^& ?( X0 L/ p! M* I- L+ m* K5 |
Dickson’s conjecture. ( G9 @3 _4 M3 W# _
digit properties.
2 L9 j) o9 F5 {* K$ uDiophantus (c. AD 200; d. 284). - \7 z8 G+ ?$ s& i( @" J8 c
Dirichlet’s theorem and primes in arithmetic series. ; R9 q+ V, p' n% }; m: c7 _5 X
primes in polynomials. & k* J& K4 @+ ?' Q* K0 U# ]) Z
distributed computing. e* _) C. r# x6 x
divisibility tests.
) @6 n$ j. v6 idivisors (factors). + p* i. e2 B7 N* b2 p- @
how many divisors? how big is d(n)?
; [ w4 ^3 K0 N" }; Q- A2 d. ]record number of divisors. 9 \. ^: L9 i; S% t( c* w* h
curiosities of d(n).
) ^9 [( k+ D) }- Y- Adivisors and congruences. # ?- H6 @# \2 O
the sum of divisors function. . S" Y1 c7 e8 f7 f" Y% |( |* b
the size of σ(n). ; Z; u1 Y- v) N- h+ ]' A, C
a recursive formula.
3 T. r/ ~# v q: }divisors and partitions. & Q/ d4 f& C. t1 ~! [
curiosities of σ(n).
) N% r- W8 i- ~& qprime factors.
2 E( a1 r) a+ {" s" ?divisor curiosities.
/ {" ?! R8 i: V. X9 A0 m, Geconomical numbers. + s6 V$ H8 {2 x! F% f& t2 j' P
Electronic Frontier Foundation.
- u& O# |7 H# N* S, velliptic curve primality proving. 6 O2 X7 q0 V* \5 ~
emirp.
6 T& y+ E3 w: {. F# cEratosthenes of Cyrene, the sieve of.
! l- W) T$ R3 Y- w6 eErd?s, Paul (1913–1996).
& d2 j5 W- t5 [' Z. H% k! U/ \& Rhis collaborators and Erd?s numbers.
0 O) {! s4 a" }errors. " t4 E2 D4 q9 h" o/ I" U" Z
Euclid (c. 330–270 BC). ) _" r" P' @8 t% m# E
unique factorization.
# `2 \: z& G6 s&Radic;2 is irrational.
, M; t6 |9 S) N5 ^: P. l0 ^2 sEuclid and the infinity of primes. ; }3 a' h7 f: n: E, p4 k
consecutive composite numbers. 5 c y) |* H5 X
primes of the form 4n +3.
4 M) y$ w1 A. i& B7 A4 @1 w& \0 L2 Ja recursive sequence. 1 t a& e% N Y) h3 A q" Y4 |
Euclid and the first perfect number.
3 ]7 n7 F2 g9 `' g5 h2 m1 K; JEuclidean algorithm. - O7 ^% x5 O) ~# Z6 E
Euler, Leonhard (1707–1783). 2 ]2 N4 f6 ~% `# o- j
Euler’s convenient numbers. 8 c7 {1 c$ H: h/ b0 |' L. Z( m: X# C
the Basel problem. ( s; ~# c9 i+ n; h$ z5 A: H" g
Euler’s constant. $ R( |' l a( T# q1 }* A
Euler and the reciprocals of the primes.
: l( K0 k+ R1 D9 S [Euler’s totient (phi) function.
2 q0 y+ m. ^+ y" \) \+ S! qCarmichael’s totient function conjecture.
7 f: ?' P/ ?6 I, [curiosities of φ(n). 1 C5 s$ @) [# y* o( E+ i' P
Euler’s quadratic.
 d( l0 }5 t, m+ W# _the Lucky Numbers of Euler.
: \4 b$ @: x: {( p/ W' tfactorial.
% T- p- v% x: t& C% s( m4 x- nfactors of factorials.
7 Z& l- H( |& A4 `6 n( i$ ]factorial primes.
1 s9 v7 Y+ k( q+ ufactorial sums.
8 F$ A" u7 j8 i. t$ M' P2 V, }factorials, double, triple . . . . 3 N+ c/ J D+ k- `, w9 B
factorization, methods of. . L" T* Z* y5 H1 ?3 S, B+ o3 t( Q
factors of particular forms.
. C- O5 w# s8 |' F& J2 P4 gFermat’s algorithm. & k0 B0 v0 x4 v [
Legendre’s method. ( q7 u5 u7 F" n0 y: y
congruences and factorization.
( \' q. t7 s$ T7 O% c' @: Zhow difficult is it to factor large numbers?
; j# J2 ~* G' c/ a; \: h/ equantum computation. , {- T x3 q: E7 h8 R
Feit-Thompson conjecture.
; s9 T; a" {- \Fermat, Pierre de (1607–1665).
' N8 G) J* I( t+ z! W0 w' F# P; W1 SFermat’s Little Theorem.
& M( p* R8 p; N& h l* } IFermat quotient. % H" c) o4 n/ b3 ], Y% I6 Z4 [. f8 M
Fermat and primes of the form x2 + y2. $ r" c0 e; D+ }) u
Fermat’s conjecture, Fermat numbers, and Fermat primes. * b n0 y8 a \5 K1 N" K5 P
Fermat factorization, from F5 to F30. ( S4 u+ u5 L6 C0 m9 X7 N
Generalized Fermat numbers. 7 `! d. k$ P3 v
Fermat’s Last Theorem.
- D+ Z8 m+ \' N& ^. w z& ~; Rthe first case of Fermat’s Last Theorem.
; k, }8 A9 b' J+ q8 o3 sWall-Sun-Sun primes. " t& Q& d8 I9 d' g& _/ r
Fermat-Catalan equation and conjecture. & Q5 D% ^4 V6 @6 z7 h
Fibonacci numbers. ( m7 b; N8 e( F- K6 p. }8 ~! K
divisibility properties.
: J# {2 D0 C; OFibonacci curiosities. G8 {8 n! O9 x. q' Y& P
édouard Lucas and the Fibonacci numbers.
5 Z7 V0 _. h4 B) _. G# TFibonacci composite sequences. : ? a9 p. n" q1 [) w
formulae for primes.
% d3 v. E0 ]$ G( @4 X3 J$ @" n) a3 uFortunate numbers and Fortune’s conjecture. 8 b0 t% P& o5 d3 h5 g
gaps between primes and composite runs. " K9 o/ P: [. X
Gauss, Johann Carl Friedrich (1777–1855). $ `1 j* j. m' _: J/ s
Gauss and the distribution of primes.
" z6 ]. z* U1 Y0 {- E4 T, K) wGaussian primes.
, {% K! ?2 W. Y" ?, m& Z$ jGauss’s circle problem.
% F8 ~& x/ s; BGilbreath’s conjecture.
, [ k3 {9 ?8 T7 \1 [GIMPS—Great Internet Mersenne Prime Search.
, D5 ^6 W! E' I& p, q3 |% EGiuga’s conjecture. 1 `) ~% V9 Q" a4 x
Giuga numbers.
0 f0 ^0 h" z+ \" b ?, \+ I% a! w8 yGoldbach’s conjecture. 6 K* T! _; N9 v; \) M- H
good primes. 1 [/ ^# @& e$ M3 {
Grimm’s problem.
- |9 X/ w6 a3 R- D8 `Hardy, G. H. (1877–1947). " ]6 d5 z+ Q3 b. o/ m6 e) i/ u; t p
Hardy-Littlewood conjectures. ( R$ w: c1 H' o; T: V: J
heuristic reasoning.
$ x. K( w$ ^3 @3 t9 ma heuristic argument by George Pólya.
+ X$ H& L3 N9 Y) v+ D9 D8 a, ^Hilbert’s 23 problems.
- t3 Z0 n6 w( z6 e6 ihome prime.
. z: ]1 z4 i$ B2 _( R7 S: ?hypothesis H. 8 o" @4 N8 ~- p, [+ }
illegal prime. m6 X" K. Z( ?, Q
inconsummate number.
8 I! ]8 u! z: T7 {$ l! x8 Finduction.
1 K( t9 P! u" g# Ujumping champion. 4 A0 d2 I6 ~' [# O3 t8 R& C
k-tuples conjecture, prime.
6 c" ?2 {2 O G; n" kknots, prime and composite. / n: h! ]7 p6 r. Z. z5 `' C
Landau, Edmund (1877–1938).
/ G) F- u5 `0 Q: t* nleft-truncatable prime. 9 e9 A2 l/ d& P
Legendre, A. M. (1752–1833). . W8 `7 r# @7 w* F! ^
Lehmer, Derrick Norman (1867–1938).
6 {( n" X8 i. m) ALehmer, Derrick Henry (1905–1991).
& a: p+ g4 i3 [5 L0 g( p' ?Linnik’s constant.
* W+ I$ S% s9 q, FLiouville, Joseph (1809–1882).
: j3 m2 l9 Z4 y" T) E+ x! ]# @Littlewood’s theorem.
$ W i3 t8 E1 ^3 v+ [the prime numbers race.
) s$ X ]" }( ^7 n- J: C- G2 \" ALucas, édouard (1842–1891). ( D( u+ j0 q1 E$ p1 T+ Y
the Lucas sequence.
' ], l% \/ y. v. w# t% Z! }3 y3 `3 vprimality testing. 3 T5 d0 X$ O ~' W' y6 m1 o0 E, ~: l
Lucas’s game of calculation. / b, ]/ B& g( l% X' G, p, Y
the Lucas-Lehmer test.
3 Q# ]+ z: s4 |1 P1 s" Klucky numbers. 3 N' L1 I/ Y, S: Y! N/ [7 a
the number of lucky numbers and primes. 4 N, ]8 [/ ?6 l$ f j* o: b3 p
“random” primes.
$ w/ E0 A9 `. L& M9 b) X$ gmagic squares. . w c' |! L1 |& ?+ O
Matijasevic and Hilbert’s 10th problem.
+ `6 ^, v3 I/ L0 C( @3 E0 |; T+ HMersenne numbers and Mersenne primes. - f' F4 k+ M+ M( Y
Mersenne numbers. 7 Y& L$ E8 l I. d6 K, Y
hunting for Mersenne primes. ( P! B+ P8 L" l( N) ~2 p
the coming of electronic computers.
0 J( z2 v. ~/ i; ] e6 t! T" TMersenne prime conjectures. 4 X; f- e# v. S
the New Mersenne conjecture. 6 L9 \6 X2 q% D# i
how many Mersenne primes?
: j& R7 b8 p4 j# VEberhart’s conjecture.
& x% [2 \' Z2 @% t) mfactors of Mersenne numbers. 1 i2 j5 f, l8 W+ B# \
Lucas-Lehmer test for Mersenne primes.
- H/ `; y; s* ~; EMertens constant.
" x5 Y: P3 _; J2 C! ~Mertens theorem. * l7 k. _( x1 F% r( I7 Z7 ~: T' p
Mills’ theorem. ! }$ d+ C- X3 f: y4 `, ~1 _
Wright’s theorem.
4 P% b- X1 Q; \3 ?mixed bag. d: q- q* x2 g) }) [) s* X* [8 w
multiplication, fast.
6 Z$ L/ T: Q5 T* v1 i! J. P! M. j1 JNiven numbers.
5 d4 k) B, v9 P6 A( W' p7 d/ p4 Dodd numbers as p + 2a2. 6 U5 z9 w# }2 {; f# j3 n' h
Opperman’s conjecture.
( W" K$ |( q3 }6 k" \. Mpalindromic primes. . U) V/ s. @$ {1 S& G
pandigital primes. . r4 m4 ~1 ~ U Z& Y3 n
Pascal’s ** and the binomial coefficients. % Q: h: ^ I2 o- R3 W: }, j8 ~
Pascal’s ** and Sierpinski’s gasket.
" `. d4 e$ p# G* @) aPascal ** curiosities. 7 \# m- g0 t, m% A
patents on prime numbers.
' L) a$ g. G0 F8 H) Q8 W$ d& kPépin’s test for Fermat numbers. 0 c2 h8 f3 X6 c5 Q2 L" l1 f, R
perfect numbers.
( F- l( y6 [# g ~5 F2 nodd perfect numbers.
0 |, `1 F$ j# x% h1 S4 Hperfect, multiply. / P. ] V+ A: t6 f8 f
permutable primes.
' s5 e; l8 r: x, Qπ, primes in the decimal expansion of.
7 h* F. K$ b, T* P/ mPocklington’s theorem. % i: y# ?, _! J0 @2 O0 F
Polignac’s conjectures.
$ a4 k1 ~5 N1 gPolignac or obstinate numbers.
) }, e! r. y. v5 D: Rpowerful numbers.
) z% @- _7 X' L( I8 ^! cprimality testing.
' p+ ^7 x+ Z7 u! _/ u9 I) Bprobabilistic methods. 2 S$ \4 ?9 |+ j7 b1 N
prime number graph. 7 v6 S; |" L ?, h( @% g4 ~; C- `8 N
prime number theorem and the prime counting function. 9 o. o/ }% P' t" R; N9 x; |
history. 8 y' `- V' q1 w: G/ D
elementary proof. ' u( A" H( w: V+ L/ a0 X1 {* y, K
record calculations. 6 G# i# z q; @+ F1 v2 y* L
estimating p(n).
1 `& c* g6 v1 e+ u; ]- Qcalculating p(n). . x) o E' V% W" N/ T8 j n! B
a curiosity.
1 Y, \! V: j* ]; e1 J4 Oprime pretender. ( A. V6 H! m8 U# {9 ]
primitive prime factor. " n, F: ~) f' R& l/ @9 ]) j1 {% r
primitive roots. m7 ^8 \( ?4 W/ t# p
Artin’s conjecture. 3 [. A" j- v- J
a curiosity.
0 {6 G7 J- m I& l2 Q. H( v, x0 E2 jprimordial.
8 {8 U# e0 I% @* ~3 P }3 \primorial primes.
, L& ~* v( M. {7 h- e- EProth’s theorem.
4 ?& b: z4 e8 h& r% spseudoperfect numbers.
) _3 V" a' e/ V# V ipseudoprimes.
0 T5 C- h: |3 } s( n2 _0 ~bases and pseudoprimes.
7 z; N" d2 N3 ^' T7 h, \pseudoprimes, strong. . U# I: \6 f" ~; n4 a+ y
public key encryption.
5 }7 l7 Y! V$ Q" s8 b6 F! F2 m7 Y- Jpyramid, prime.
0 `3 Q! J. ^- u g: g5 ?Pythagorean **s, prime.
- L8 j' _6 |, H0 r* uquadratic residues. 4 {5 h$ U* N* z+ l: j z. y
residual curiosities. % x& ]# N, v ?' H
polynomial congruences.
7 l ^0 J% L% h* j) j+ hquadratic reciprocity, law of.
, u9 I1 ?# W, s! xEuler’s criterion. + |, O8 h( |! ]1 ?5 \' w; r
Ramanujan, Srinivasa (1887–1920). 1 j/ t5 x( |; V2 Z7 }
highly composite numbers.
, H5 V/ M" W, g% k4 brandomness, of primes.
' ~- h( N- L/ e( A. l4 gVon Sternach and a prime random walk. % a9 X3 z% j8 U7 k) ^+ Q
record primes. 1 i, l% { B2 I, w3 P' Y8 t) q7 r
some records. & q ?2 J+ C! ~
repunits, prime.
+ d; v' V r2 VRhonda numbers. - W8 Y4 B5 t Q6 p4 [
Riemann hypothesis.
$ m. n3 }7 O9 I2 ?9 s }the Farey sequence and the Riemann hypothesis. ) h7 y' N4 d) k% c0 C0 P
the Riemann hypothesis and σ(n), the sum of divisors function. " s! u& j7 m" H) Z3 z( o, w
squarefree and blue and red numbers. 7 R% o* |( ?7 V) n( ~& o' s
the Mertens conjecture.
$ z" M6 X; ]: m ^Riemann hypothesis curiosities. ( m) U0 v- p! j
Riesel number. / k* H4 x Z5 n6 b: _8 M
right-truncatable prime.
* ^5 `+ d* i0 i! Z- WRSA algorithm.
$ f3 t& m$ A) B/ K, r3 _/ s! ~Martin Gardner’s challenge. : q4 j6 W7 p- Z+ Y: H
RSA Factoring Challenge, the New. 0 _7 d" n5 k y: e) y$ }) e5 e% h& l
Ruth-Aaron numbers. 8 _& \" b' [4 d9 @ r/ W$ ^
Scherk’s conjecture.
6 [: a5 i' a+ Lsemi-primes.
+ F( h: c: ~4 U- J**y primes. 6 n% [% t. e9 m2 p! J9 W
Shank’s conjecture.
, w% i+ P w7 BSiamese primes. - \0 i; W" [ R8 N5 B- _2 u+ I. f
Sierpinski numbers.
Sierpinski strings.
* ~$ S% h; {9 h: Z; _( ISierpinski’s quadratic. 4 T0 J `! d, h f$ y5 B6 M p
Sierpinski’s φ(n) conjecture.
3 V4 q4 N6 {0 C4 h0 PSloane’s On-Line Encyclopedia of Integer Sequences.
* [6 F- [+ s' ?/ HSmith numbers. " p2 h, \" Q0 N3 n
Smith brothers.
) |# S9 w/ O) }+ f( Csmooth numbers.
$ t8 C7 w, a8 j2 a( d, T0 A$ VSophie Germain primes. * v i# H0 h0 J* s& T
safe primes.
) S% v; {) z7 H9 r# }- csquarefree numbers.
# L" Z% v6 Z m& V4 ~: T/ HStern prime.
5 g( h2 }0 l. i- _strong law of small numbers. % ]& d3 C: S* f6 y5 r7 A, b) u
triangular numbers. ~( J% p9 g" S- ]! R3 c; r7 r
trivia.
1 Y, O3 @+ I7 j3 J# V/ o/ l, mtwin primes.
- l/ d$ B! V8 R' c0 w' `twin curiosities.
% U" \- u+ K' Q5 V; Y2 [Ulam spiral. 0 B, r! I- O' [; _: t3 M
unitary divisors.
6 p( _% J+ o" p: B+ j, ~unitary perfect. ' K5 O5 F2 g+ w ^
untouchable numbers.
( k6 g8 J1 j' H; Y9 mweird numbers. ( p4 R/ o5 E! N: K2 u. B( p9 m# |
Wieferich primes. ( b! W# t! m% A$ s7 J3 I/ z, V
Wilson’s theorem. 3 M) W% D6 p; E$ f" V
twin primes. ) I0 t, R1 ^) x% h: |
Wilson primes. ( D6 ~; D" V; C
Wolstenholme’s numbers, and theorems.
# [& Y9 O R' O" H; M$ N2 `5 |/ cmore factors of Wolstenholme numbers.
 i- P J t. RWoodall primes.
# t5 t* h! h7 `6 x, Yzeta mysteries: the quantum connection.