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数字的奇妙：素数

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Entries A to Z. # P" e2 a! P# _1 v7 K+ G" [" l
abc conjecture.
. P4 n: ^! p( j8 `; U6 N- g- gabundant number.
% T' v; L; C4 z) f: _AKS algorithm for primality testing.
# E$ v5 D. L8 c* n+ M6 S( Baliquot sequences (sociable chains). 2 W- I; G( e: W. k; G, g
almost-primes.
+ E9 D% m' u2 s R/ h: ^/ n: vamicable numbers. , ~ `& h* R8 j* T$ @
amicable curiosities. , m3 i2 ^$ p) v; v3 L5 M
Andrica’s conjecture. : R V& p/ s2 s5 ^
arithmetic progressions, of primes.
' ?( {, S& X& CAurifeuillian factorization. 5 C7 d. {2 J8 Y. c, C1 F
average prime. " ^* e( n( J8 M) |% t5 {
Bang’s theorem.
- l( X; Z8 F4 O, j0 U- V+ cBateman’s conjecture.
# J* y, o% b5 R* \Beal’s conjecture, and prize. ) D* J. l+ N8 N: X2 B7 n
Benford’s law. 4 a; q: _6 N- I# R
Bernoulli numbers. ; K7 s9 P* E, X. k" h& ~2 y
Bernoulli number curiosities.
 d" ?. J+ S/ O$ K! @Bertrand’s postulate. , B! b4 ?/ w: l4 f7 C. t
Bonse’s inequality. 9 A* ^8 _. N- j1 u8 d$ V
Brier numbers. : `! X! ]% _; b0 ]$ M6 k
Brocard’s conjecture. / F( G' }% o. G; B, T0 V3 t1 {
Brun’s constant. 6 a. v* Q' t# A: E7 v
Buss’s function. + [% e0 U/ q" V) L7 Z9 g+ ^8 v
Carmichael numbers.
, M' Q- C1 R1 H OCatalan’s conjecture.
; W/ Z& r+ ], P: _Catalan’s Mersenne conjecture.
( t* p B9 M5 G, {4 v4 CChampernowne’s constant.
( f0 ]& t9 F m7 w) ?4 k, g Hchampion numbers.
4 N: F( z, x+ s% p+ ?' P4 j+ IChinese remainder theorem.
( `+ n7 j4 V" i/ d; Tcicadas and prime periods.
" l% T2 M1 u: a* j! o" [6 e( kcircle, prime. ' a6 [7 j, d) e5 u& O( e: q. r4 L
circular prime. 0 y; H0 j: k5 f9 z& M
Clay prizes, the.
# H) L6 Q" l. J7 v+ scompositorial. \ f5 F& D$ c8 R- t; o! e* T
concatenation of primes. 4 {/ h$ N5 h( S# U( _9 f$ i9 T$ ^
conjectures. . m8 k- R. ` U' e
consecutive integer sequence. # H! V5 H+ G4 I8 c
consecutive numbers.
8 I2 p% P4 ]$ @4 V( xconsecutive primes, sums of. ) w& X) g% s( N' m7 c! ?- R0 R: u: g8 k
Conway’s prime-producing machine.
+ ^/ Z+ o3 J4 ^! gcousin primes.
5 A( q& v6 i, P1 qCullen primes.
) o6 x' x" p" w1 h/ w! Z: O, }$ ICunningham project. . i7 ^; e* S5 [, ]
Cunningham chains. 3 d; c% ~9 B% f, v7 G: Y# e* v, ~
decimals, recurring (periodic).
1 U6 D4 m* i3 V7 i, sthe period of 1/13.
2 C8 v6 c n2 {4 @cyclic numbers. ! G: e: i( V6 p3 A8 x
Artin’s conjecture. / c; E0 ]5 O4 Z) {" y
the repunit connection. - l5 Q3 ?* i" Z! b( m& v& H
magic squares.
; j) O$ c, s% l, P( kdeficient number.
: L+ ^/ {& n& b, } I/ n" R& f7 H, Tdeletable and truncatable primes. . @* M2 A, t" T2 M
Demlo numbers. ; i; r0 ~$ _ }* I) [
descriptive primes.
# {! P3 K, C0 m$ b7 _7 W; X3 ODickson’s conjecture.
- ]; H+ D# _: `' `2 K* ]digit properties.
# K5 I3 H! r! [$ G, s* _Diophantus (c. AD 200; d. 284).
; p0 S2 |$ z/ fDirichlet’s theorem and primes in arithmetic series. % }+ I+ z3 Q' [
primes in polynomials. # D! X* h5 q( b1 {$ w" @2 n
distributed computing.
) V0 o& y+ A& qdivisibility tests. " N- T& f4 R5 D' i# i9 l. u, Y
divisors (factors). 2 E& w5 Z; F" b4 P* r+ [5 d' J
how many divisors? how big is d(n)?
8 W$ K9 y- ]" \) E8 [/ @( i1 @* k3 Irecord number of divisors. 7 ~6 ^& r' g) Y: c; }
curiosities of d(n).
" [3 W( L7 R" I* t1 _- @7 \divisors and congruences.
 \0 l& f( M+ N7 K* Gthe sum of divisors function.
/ r6 s; H0 i" S2 k4 Xthe size of σ(n).
- [& e; Y) K6 X% `7 @! `" la recursive formula. ! a( V* ?6 }) e0 c+ Q: ]) L* `6 g9 C: d
divisors and partitions. 5 p0 C% t+ D4 g; r. T
curiosities of σ(n). % ?4 T( }$ ^0 [
prime factors.
) x0 ]' l# m2 j+ N2 R* xdivisor curiosities.
( t% {) C8 M7 {) e* |economical numbers. + c+ M/ Z: \% z* {2 L
Electronic Frontier Foundation.
+ B2 `* D, L" g" @* }elliptic curve primality proving.
& a. p4 T$ k' T2 Zemirp.
 ~2 n. D# b6 a) e6 i" uEratosthenes of Cyrene, the sieve of.
* G( Q4 B4 h! B' wErd?s, Paul (1913–1996).
( {/ @0 O" x0 {" chis collaborators and Erd?s numbers.
8 z- B9 r- o+ V" g, l) s' U; J' xerrors.
. e" z: E) h" ~/ u3 a' ~0 [8 SEuclid (c. 330–270 BC). & [9 d7 K$ p6 ~
unique factorization.
- _% a, y1 a* q) a+ ?* V9 O5 _&Radic;2 is irrational. 9 ^- P) }, u) z" Z
Euclid and the infinity of primes. 1 w: l* P/ s ], z1 C
consecutive composite numbers. ( R; Q O7 ]9 G( E& J6 l, z2 b
primes of the form 4n +3. . F" T7 D( i3 o2 p5 o) _2 D
a recursive sequence.
1 B: B5 ^! ]- U, M6 v0 E. }Euclid and the first perfect number. ) d1 a8 I7 g; P
Euclidean algorithm. ]4 T6 O) b; g% [" `8 B
Euler, Leonhard (1707–1783). " Q" K X [) n
Euler’s convenient numbers.
3 t. ~1 E! S5 J+ ?/ Uthe Basel problem.
/ C8 W! i7 `* g3 u9 V. `4 uEuler’s constant. ( h5 y8 E' y' P* v
Euler and the reciprocals of the primes.
& V8 `# \1 m- T) I( OEuler’s totient (phi) function.
& [, o4 j8 W; w' g& NCarmichael’s totient function conjecture. , c, N9 v3 q1 J d* `: Z; J
curiosities of φ(n).
0 y) D5 t M& N) p x/ g- ]Euler’s quadratic. 9 n0 z u2 g: g7 y! C
the Lucky Numbers of Euler. - k. @( y! X9 o. h) N/ S
factorial. ) K% l* J! F8 R$ S$ N
factors of factorials.
+ \) K6 t. ]* W+ \ c2 nfactorial primes.
$ I; ?( k1 h4 D9 Wfactorial sums.
8 N' c! _1 @, E$ [0 Bfactorials, double, triple . . . .
! f, j$ e! x$ A* I) A* F6 D! dfactorization, methods of.
4 Y- s) O- V; l1 o# ffactors of particular forms.
- N1 T; }* k ?) \8 sFermat’s algorithm. : T/ G) ?$ J: s8 w! q- {
Legendre’s method. 7 C: R1 g: }( y
congruences and factorization. ) Y7 |: U% ~+ Z: h9 P+ a
how difficult is it to factor large numbers? ) `/ N( ~& G2 Q; i; T
quantum computation.
7 X; K2 j' ?6 i+ p3 O! r0 |$ DFeit-Thompson conjecture. ) O7 [( c% G! B/ j
Fermat, Pierre de (1607–1665).
+ Y) `, D/ E4 ? d! N2 |$ zFermat’s Little Theorem.
* N4 _1 u3 ^! L' s; ~9 E5 fFermat quotient. ! z- d+ m2 L g- F* u0 }1 \
Fermat and primes of the form x2 + y2. ) u& @) r* g6 Z H1 s
Fermat’s conjecture, Fermat numbers, and Fermat primes. 1 W: M" c5 B3 z. T( v9 i5 n
Fermat factorization, from F5 to F30. 6 s2 o7 \1 o* P* V
Generalized Fermat numbers.
Fermat’s Last Theorem. 9 L/ F; ?- q' P& ^! w8 L3 c5 g
the first case of Fermat’s Last Theorem.
% I: ^+ ~8 |) e6 x/ h8 M) H: {/ RWall-Sun-Sun primes. ; J' I7 D N0 u& A
Fermat-Catalan equation and conjecture.
5 k6 w, \) V6 J6 DFibonacci numbers. 6 x& p( B# v# g1 I& V/ X6 C
divisibility properties. 4 I2 {6 z7 n. k; y
Fibonacci curiosities.
( {, d$ i- {6 s- k0 R4 médouard Lucas and the Fibonacci numbers.
& r& ^- }8 W* g: X; w! s4 F. \. FFibonacci composite sequences. ; r1 |% H* C. C' K( f. M
formulae for primes.
0 r4 |* \6 c4 g% N: ~. Z* n& bFortunate numbers and Fortune’s conjecture. , V# B- ^# Q- x+ F
gaps between primes and composite runs. - f& N* i( ?3 |! y* X2 o, H
Gauss, Johann Carl Friedrich (1777–1855).
) q1 z+ T( y, i8 k8 J7 k! _Gauss and the distribution of primes.
( G% ]$ c4 Y+ v1 I& }" U& tGaussian primes. / @. o4 f2 j) ^$ X3 Q
Gauss’s circle problem. $ I+ W# S: p; X( Y3 E- D4 N
Gilbreath’s conjecture.
: S* {- j) O$ tGIMPS—Great Internet Mersenne Prime Search.
6 x7 t. J# Z) M) M6 I8 ?: fGiuga’s conjecture. / a8 A$ c& |/ V5 s1 X( D( m
Giuga numbers. 3 ?: h! o$ b3 R2 F
Goldbach’s conjecture. 7 a7 N( B7 d& P/ _$ D
good primes.
+ x# \7 W" Z" B4 ^/ VGrimm’s problem.
$ D" d0 z' _7 E8 @$ ~; [Hardy, G. H. (1877–1947).
3 e& T2 f w5 y* ?& Z+ H' iHardy-Littlewood conjectures.
$ \1 ?) O0 @! I$ j6 Hheuristic reasoning. ; M+ Z) ^. I% C* R( ]& L
a heuristic argument by George Pólya.
) g% B9 @4 Q; _0 C7 L! ? E! H! `Hilbert’s 23 problems. # c- g+ W$ b; |3 ~1 v9 N/ p
home prime. 7 o) Y+ _, B8 P! T. x
hypothesis H.
( s; }5 }7 N, n9 p8 ~& B! _1 t Millegal prime.
$ Y) \4 w% P# g# hinconsummate number. 0 {% p% A# L2 v4 c. N: C' e) Z. P
induction.
7 | C6 u( m# n/ G+ K* Ujumping champion.
# ]' e% F0 L. g9 Z7 R$ V, b9 p- Yk-tuples conjecture, prime. : i0 S5 `! g$ d/ h5 w3 |& r4 r
knots, prime and composite. * V3 M, `! q$ L+ V
Landau, Edmund (1877–1938).
; \8 \4 a ^# j7 x8 K4 o ?" sleft-truncatable prime.
3 y# K( }8 |: T& L1 pLegendre, A. M. (1752–1833).
" m9 M4 i( S! e6 ~% n6 M3 I/ _Lehmer, Derrick Norman (1867–1938).
& J7 b0 d3 L* r! a8 WLehmer, Derrick Henry (1905–1991). / a. G# j3 z& E. o
Linnik’s constant. 9 x8 y! d7 i6 j6 c h3 Z
Liouville, Joseph (1809–1882).
Littlewood’s theorem.
, }" F2 X+ d6 ^, e" I5 Z1 pthe prime numbers race. ! X1 N8 F" d% u$ v
Lucas, édouard (1842–1891).
: a9 r5 x5 H, e8 y! H. Bthe Lucas sequence.
 z( T I3 u: `) x4 a, Uprimality testing.
 T B$ S% Y! _: H% CLucas’s game of calculation. * n9 x! i# A# L% d# x
the Lucas-Lehmer test. ' ?1 U5 }; ]+ T
lucky numbers.
- ^$ b" @8 o% A6 Y3 {- y5 Tthe number of lucky numbers and primes. 7 `, Z0 ~1 U6 R
“random” primes.
. y+ ~- V0 F2 q7 o' k; x% rmagic squares.
4 O* b9 }: l+ ^1 s: k# JMatijasevic and Hilbert’s 10th problem. 0 H0 d4 u6 k4 T4 S1 {) t
Mersenne numbers and Mersenne primes.
3 l _: |+ d) d. {Mersenne numbers. * O$ j3 N% L: E: H: X
hunting for Mersenne primes. - ? N( C( F1 _1 ~* |2 R& a9 c8 N
the coming of electronic computers.
& [5 N* H, s5 o6 ^! ZMersenne prime conjectures. 9 h2 s0 n) J3 P3 M# L
the New Mersenne conjecture.
- g. c) W& }% W+ h# g1 ~9 chow many Mersenne primes?
4 A/ |0 x+ J F& [3 J/ uEberhart’s conjecture.
3 l' A! h, w, ], `. _factors of Mersenne numbers.
1 ^* z* a. D9 J# b' b8 V9 gLucas-Lehmer test for Mersenne primes. 3 t7 a# ?* l) k# V$ ~6 n+ A
Mertens constant.
0 q6 h3 O8 z1 p7 o7 U7 r: LMertens theorem. s# W1 Z1 p: [% [* }1 u# w
Mills’ theorem. - H: [2 a1 C& q
Wright’s theorem. * I% a9 j8 j5 W# Y8 q
mixed bag.
- u& d0 z3 u. W+ Omultiplication, fast. ! Q V. _- h3 ?8 w" q4 [$ v
Niven numbers. 3 Z2 ~ ]. C5 Y& Y
odd numbers as p + 2a2. % g' g. E9 W G" a# h
Opperman’s conjecture. ( D( I+ r! u, @) \
palindromic primes.
& h8 W8 v7 r( o2 c: Lpandigital primes. , @) v! a) D4 I0 M. \3 X- ?5 v
Pascal’s ** and the binomial coefficients. ' }) J, U8 |, b; @! n, d
Pascal’s ** and Sierpinski’s gasket. |4 n* K$ w* S
Pascal ** curiosities.
6 C0 u5 @/ q( D; ~0 `patents on prime numbers. 5 G) ]3 `/ ]0 i4 C6 B
Pépin’s test for Fermat numbers.
& Q0 {) x2 Y% l$ e: rperfect numbers. 1 r9 k, w" \3 c2 h: H: s8 T
odd perfect numbers.
( e: N6 q/ L$ A0 Dperfect, multiply.
' F- Z0 J: C$ \. h" ]permutable primes. , E0 ~/ o- m4 V4 |
π, primes in the decimal expansion of. 4 s6 X4 n. t; ?7 _' {6 |
Pocklington’s theorem.
& |3 T3 B2 ~ S7 VPolignac’s conjectures. " G3 L1 J/ e, p3 d, y
Polignac or obstinate numbers. 7 _1 d, T, I! e4 n ^% e' ^
powerful numbers.
7 E) x# D: h! V' ~5 `4 dprimality testing.
( T- b; p& i1 h" X Jprobabilistic methods. ) }/ Z" A+ d% M
prime number graph. 9 }% A' [9 |, S9 U5 s2 W/ e: n
prime number theorem and the prime counting function. % d8 M0 T! G; B- X1 k
history.
7 \- e! b% v- [" M, v. zelementary proof.
+ D. _7 d, l i! u; y- R: Hrecord calculations. % y; Z& h" `) y( s6 V7 ~$ p
estimating p(n). * K p% f% z! W0 ~$ m, B
calculating p(n).
1 _3 \ l& I) Y- G* Sa curiosity.
6 t9 v5 W! g- J' Fprime pretender.
8 e( x& C; r3 [4 G0 s$ fprimitive prime factor. 4 A2 v! E; O4 X7 @3 n; _
primitive roots.
1 z+ v! ~3 k; ?, S7 {" }Artin’s conjecture.
: N% p% m. n8 na curiosity. / G( j' J; d+ L- F/ M* g0 Y8 X
primordial.
0 u S2 p' @' R# fprimorial primes.
2 E: j0 a% f4 P# _" OProth’s theorem.
 \; S7 D" m5 [. ~( E( Kpseudoperfect numbers. ; \+ |/ Z2 U; ]; O( X8 X9 g
pseudoprimes. / k1 n) ?9 z- h( m% @. q
bases and pseudoprimes.
: J6 P; H& ?7 r( A1 lpseudoprimes, strong. 3 S4 R; `( Y0 v% a1 C* {
public key encryption. * f, p- c2 q- R, f
pyramid, prime. 4 l- k1 Z2 }7 `2 ?. i0 V
Pythagorean **s, prime. # h% j! o& h$ H2 E# l$ p4 m8 r! a
quadratic residues.
/ h8 b4 Z0 Q' J8 jresidual curiosities.
9 e$ F, C; a/ Z( X; Wpolynomial congruences. 7 ^6 V) C' ~4 l9 ?9 K7 U* s" \
quadratic reciprocity, law of.
/ A: [) s2 `. V, l9 FEuler’s criterion. : ` c' n4 Q3 O5 y
Ramanujan, Srinivasa (1887–1920). 4 {9 y, o) _' I" \ G3 n- I
highly composite numbers.
8 g$ J/ ~- |0 V( ]9 H/ \randomness, of primes.
4 _8 H1 g/ z l6 LVon Sternach and a prime random walk. `, l0 q# v% J1 A9 ?
record primes.
; \/ I( Y2 h: g0 ?/ esome records. ! o3 n$ [5 B6 c5 |% a8 z) F* _
repunits, prime. 1 i( D- M8 n9 w# X# S ?; |% Q. P
Rhonda numbers.
4 K" R3 Z- O( uRiemann hypothesis.
/ s5 n4 F$ Z3 q( C w' e0 Vthe Farey sequence and the Riemann hypothesis. O( V; x/ |, E8 x+ j: F5 {5 T0 h; x+ e
the Riemann hypothesis and σ(n), the sum of divisors function.
9 O0 |$ R% K4 b7 }, S; ^squarefree and blue and red numbers. 8 V. ^6 J' s( Z% N
the Mertens conjecture. 1 f" f7 P0 M$ }5 c$ F, i
Riemann hypothesis curiosities.
) ?, ^5 `0 f P. r3 ORiesel number. # d4 u) N2 r& C- K/ C; d6 e
right-truncatable prime. " @: A% p6 r8 a
RSA algorithm.
& @! F% B: I. f; \Martin Gardner’s challenge. 2 V% m: L6 }% p8 {
RSA Factoring Challenge, the New. , [- k% W6 c- W" E1 Z* u# u
Ruth-Aaron numbers.
 a1 L8 p% \7 U$ [Scherk’s conjecture. ! d' x3 m/ d% P, k/ I
semi-primes.
- i2 ~1 I2 d' X; V- A u( t4 ?1 L**y primes.
& e' ] z/ @0 A r% x6 s3 IShank’s conjecture.
; w. w6 w" F* B; k) o. dSiamese primes.
2 y1 a. ]* e3 g% n4 H- A2 vSierpinski numbers. " p0 _4 p6 l' R8 ^7 K0 \ O
Sierpinski strings. $ X: r5 I4 G/ w" k; i p0 Q
Sierpinski’s quadratic. 9 j# u+ T4 D# ^1 [1 h% ]# G% a
Sierpinski’s φ(n) conjecture. ' R# m* z# o) l j
Sloane’s On-Line Encyclopedia of Integer Sequences. & K# W0 m: B5 q# ^( E8 p5 m
Smith numbers. ) P- E! Q- d8 u- J- `9 a
Smith brothers.
3 q5 {+ o4 L5 R7 k- X. i6 T) `5 rsmooth numbers. ! f$ ^* m8 r3 ?6 B
Sophie Germain primes.
" l, y: s3 F3 c1 h- O/ M( Qsafe primes. # t$ B$ X. ]- t9 l4 _
squarefree numbers.
4 ~7 ~2 Y6 ?; |! T) @+ RStern prime.
; I8 C7 }' k2 J) R Y) m A# j; jstrong law of small numbers.
8 E' V7 z/ ~" R# Ntriangular numbers.
6 D+ ~) X6 T5 v9 S6 P E# _trivia.
, q! q0 {' c& Y8 Vtwin primes. ; `' S: Z3 X( F% W N
twin curiosities. ; l4 x& p* Y' C6 ?. _
Ulam spiral.
7 M* v3 Y! G3 a9 f$ y* \( J% Punitary divisors. ) o+ s$ C2 }1 O- t8 z: `4 n
unitary perfect.
/ s& Q$ {0 M( O! F+ ?4 [: Puntouchable numbers.
2 V! |( p# c; Bweird numbers.
, Y4 M. }; {- E; kWieferich primes.
 O3 ]* x2 T1 b8 n6 l. h* BWilson’s theorem. A$ V$ `5 F: h7 }. s, j! a: L3 W
twin primes. 5 M- ]" C6 C* E t* `! x
Wilson primes. ; w0 t6 W/ `5 S+ G4 I
Wolstenholme’s numbers, and theorems.
5 x% p- d% T# w/ F( D0 H& {more factors of Wolstenholme numbers. 0 J# T% b% A- f6 X* [4 ~
Woodall primes. ( c3 p' n9 f& L4 Q
zeta mysteries: the quantum connection.