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数字的奇妙：素数

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Entries A to Z. + a# x; k) C8 B+ k$ u7 k7 { C. b
abc conjecture. ( e( |1 d* V, V& P5 R3 W! s
abundant number. 3 q4 X6 Y; G& M
AKS algorithm for primality testing.
. s2 @7 X6 ?' j, o$ w2 c$ [aliquot sequences (sociable chains). + T! l5 b1 D! }) s! O
almost-primes. W8 p+ b& ?; }" i
amicable numbers.
6 @( G I* d7 i; \amicable curiosities.
3 M% l2 L5 \* B; F; p; k- T1 NAndrica’s conjecture.
' ^; b5 T! G C* a0 b. K3 S- rarithmetic progressions, of primes. / X9 j6 a4 L- B U
Aurifeuillian factorization. ; O3 K% g% r2 v% W& D- @7 q
average prime. * N& E( O Y& ~: t7 l
Bang’s theorem.
( B6 K% G2 x4 ^# r7 r9 H2 aBateman’s conjecture. 5 }2 @" f0 ]+ X( k2 d4 H
Beal’s conjecture, and prize.
$ V' z; ^$ [! o9 cBenford’s law.
% A, O) `+ ^# Q/ n5 ~Bernoulli numbers.
% c' `; F/ b, {( J# a, GBernoulli number curiosities.
* w0 D- H4 q. R) [" f4 MBertrand’s postulate. , r! a4 o" b' ]8 C) f: p
Bonse’s inequality. 3 {3 V3 l3 }. {5 ?) Z4 b% f
Brier numbers.
4 {# D) a" j$ T1 w4 U2 YBrocard’s conjecture.
- Z: T& g1 l) Z+ [Brun’s constant. ) y) Y9 w C- ^2 G
Buss’s function.
: D- s+ c5 M. ?1 z1 e: S4 OCarmichael numbers.
% w8 K; T" `5 x3 D: {Catalan’s conjecture. 9 ^% S& z# F p* {4 g3 ~+ @
Catalan’s Mersenne conjecture. + ?# a9 T9 l S) ?$ _4 [) ~/ c# X- {
Champernowne’s constant. 2 W- L2 v, h) F$ i; m
champion numbers. - c' _) S) I0 K- P5 A% p! t
Chinese remainder theorem. - |# N. y. A( T" P
cicadas and prime periods.
' W/ n L0 ?8 [ }: f6 F2 Ycircle, prime.
: t9 P( Z3 R' J0 L. K0 e- g4 Zcircular prime. 1 P5 O4 a- D, J- o3 D
Clay prizes, the. * O/ O( W/ O: ?, u8 n
compositorial. / Y# T* T" m! K( d; I0 m
concatenation of primes. , N3 J: N) T" U1 b9 J) d0 k8 h- |
conjectures.
) C$ m, A& w$ S4 @9 ]. S9 Cconsecutive integer sequence. / u% o! F3 W8 e, `
consecutive numbers.
; M) z: N1 I4 \- k. Bconsecutive primes, sums of.
2 G t2 s7 h3 ^Conway’s prime-producing machine.
; u C* h) {2 c& Z5 [cousin primes. + {4 R. L' y# |! H9 P8 K
Cullen primes. / l3 k: O8 w+ l: r* h+ ]0 i3 h
Cunningham project. ! h$ q+ n/ h8 z* ~$ E# _
Cunningham chains.
9 c' S( H2 E3 J Zdecimals, recurring (periodic). 9 D g6 A1 s% x' |5 c9 \. k
the period of 1/13.
; v) E4 z# y% V3 f' C+ ~0 x% ]- w4 m; Vcyclic numbers. 7 R& T% I8 J1 c; ~; R: ^: v' O6 L
Artin’s conjecture.
/ ]% y7 z/ ?0 x, s, N/ gthe repunit connection.
1 s8 U5 X) R8 W8 K# @magic squares. : _# w9 O- ?/ b6 W' o
deficient number.
2 }0 r( L0 j2 D3 g" h/ mdeletable and truncatable primes.
* B: }) {5 v5 BDemlo numbers.
6 Q0 h e) F4 ]1 Ddescriptive primes.
8 ~2 U" U i' x: ]- a6 mDickson’s conjecture. ' w: e7 x' w3 X( }
digit properties. / N' V& [2 M) Q* \5 l
Diophantus (c. AD 200; d. 284). 0 D; [! Q s, _
Dirichlet’s theorem and primes in arithmetic series.
1 p8 b5 E! s. @4 Vprimes in polynomials.
distributed computing.
7 Q+ q1 c* T: X7 _* {divisibility tests. * w( _9 {* \( t: S4 N
divisors (factors).
: d1 u' d( x$ e( U+ dhow many divisors? how big is d(n)? + `7 R; K5 }! F, v( z. |' J
record number of divisors. - [. x' b1 [$ q( C) c% T1 w
curiosities of d(n). ; h& N2 w) u% N& n
divisors and congruences. 6 _% ]) f$ ~7 q1 R, @3 U" O
the sum of divisors function. 8 d! j- p! O6 t% n7 k( M9 }! s
the size of σ(n). 5 s2 J$ Y1 z7 S2 Z H
a recursive formula.
; H! W% K5 W$ ~2 Mdivisors and partitions. * _* h( O5 a! E# s$ M1 B/ x
curiosities of σ(n).
6 b; b/ l! }3 l5 kprime factors.
' d' I. X. U/ f7 V: n6 udivisor curiosities. ' D9 j; k2 E) g2 @
economical numbers. * v- g" X& v- F
Electronic Frontier Foundation.
8 `0 h$ @ d( n1 i. k& N0 Celliptic curve primality proving. ! C3 B! x, \3 p" S2 Q* l6 x0 R! ^
emirp.
# Z$ H' h B% @Eratosthenes of Cyrene, the sieve of. / R: C7 i E/ `: ` ^% L# W
Erd?s, Paul (1913–1996).
7 T7 A1 ?& \- N; O( c# P7 Lhis collaborators and Erd?s numbers.
) o5 h! A9 H" G \8 p; X7 c) Gerrors. ) p a) e6 P$ Q" s
Euclid (c. 330–270 BC).
& s, O% n9 N" F- _unique factorization. + \6 u5 U3 z2 W: s. x, {/ R
&Radic;2 is irrational.
4 x# `" d9 k0 A) h0 v! ^0 IEuclid and the infinity of primes.
0 g, ^3 o0 L- }8 l/ P, kconsecutive composite numbers. 3 y" g% D7 L) k1 ~3 x
primes of the form 4n +3.
% W' [' y8 ` y; R( La recursive sequence. 3 g3 P1 w- D* U. u2 R& H: O
Euclid and the first perfect number.
& M+ ]9 r. j$ S$ L) U+ a- AEuclidean algorithm.
% }; t5 i. ~) p5 H5 LEuler, Leonhard (1707–1783). 5 l4 e R1 h# N; _+ o# a' J9 s, W% P
Euler’s convenient numbers.
) w8 x) U! A/ M3 ~2 [9 l( [the Basel problem.
8 J" _4 [, X. Y+ u* G8 S7 x4 cEuler’s constant. : o% X* P% \0 s
Euler and the reciprocals of the primes.
7 }0 q+ n) K) m7 B9 IEuler’s totient (phi) function. # `- u9 e' L4 y7 O
Carmichael’s totient function conjecture. 9 |" K0 _; d4 @: J9 K# }
curiosities of φ(n).
% o: [5 ~9 a9 dEuler’s quadratic. " a) |3 N5 [1 j* `2 C
the Lucky Numbers of Euler.
( o1 ^% @3 W* C3 O( Nfactorial.
, c* v8 S, j. l7 afactors of factorials. ( _( L- b0 A9 f' q9 `" O; d0 b
factorial primes. 2 O: I3 I7 V1 s: J$ d" e+ u
factorial sums.
# }# J$ _- p; I) _6 J" @$ U1 Lfactorials, double, triple . . . . L3 `5 A$ a1 ]# `/ Y$ B0 u$ R3 d! B6 a
factorization, methods of.
% ~1 _! ]3 a7 t, k4 F) c, G5 F% kfactors of particular forms.
, @. w! G8 \2 v. ~; g9 LFermat’s algorithm.
% L! v/ `, D, Z: C4 W0 NLegendre’s method.
$ E- F" W' K4 G b2 ycongruences and factorization. % s$ u" z; L6 ?
how difficult is it to factor large numbers?
quantum computation.
# y! m; ?% H: i- q. |Feit-Thompson conjecture. / i% p5 W) W' V1 ]+ j" H$ j6 E
Fermat, Pierre de (1607–1665).
9 T+ ]5 D0 T: K5 F* u% E3 GFermat’s Little Theorem. ' @& B9 H# t! k" c9 a
Fermat quotient.
, E* Z- r9 B) J! v, e! g9 CFermat and primes of the form x2 + y2. * H4 o* @$ w: L3 k* e9 w! v- M& u
Fermat’s conjecture, Fermat numbers, and Fermat primes. - @ A h# r2 t
Fermat factorization, from F5 to F30. 1 z* P3 q& n, K" H
Generalized Fermat numbers. # u* b) Z5 M: c ]5 X
Fermat’s Last Theorem.
1 P9 Z* e: H. z2 A, v1 hthe first case of Fermat’s Last Theorem.
2 ?* O" n: o w$ \" LWall-Sun-Sun primes.
6 v, C# l" U% C7 i! s- b, L) vFermat-Catalan equation and conjecture. . p; r' x8 K; W2 e2 d# R
Fibonacci numbers.
0 P7 {. g- q) y5 {2 c5 pdivisibility properties.
( p; I+ J! L" A0 X- W! }) ?% JFibonacci curiosities. 5 m0 A& |- I; N6 P' z
édouard Lucas and the Fibonacci numbers. " n: G! {) s5 j' I5 r' N' G
Fibonacci composite sequences. % v i, [* J4 g
formulae for primes.
* K$ x. _0 ^9 _; ZFortunate numbers and Fortune’s conjecture. " n4 j, _$ I" T* `2 r
gaps between primes and composite runs. * d# p! Z4 {1 u. B x( I
Gauss, Johann Carl Friedrich (1777–1855). 5 o1 W7 c+ J# L1 N5 Z/ C# F0 Z5 F
Gauss and the distribution of primes.
0 n9 `9 i0 G3 n7 H. z+ A CGaussian primes.
* d) @$ w2 h" i3 @" E3 r* a+ [- Y. LGauss’s circle problem. g5 F" _/ ]7 f' Q' l' t6 y8 X: b
Gilbreath’s conjecture. % P# P' b% l+ d0 q8 z" _6 K- g
GIMPS—Great Internet Mersenne Prime Search.
. \" p2 T2 r+ `9 G2 f* ]% sGiuga’s conjecture.
; h( Q& G) n; g: R5 AGiuga numbers. 7 \' c6 p7 e' }. D0 {( T; R* I
Goldbach’s conjecture. ! S4 x9 ]$ O; a# K& c& B+ x
good primes. # z7 q A/ T) s( j
Grimm’s problem.
3 ^4 g1 Q2 m1 U! u6 OHardy, G. H. (1877–1947). & [, ?- ~+ s2 l7 _
Hardy-Littlewood conjectures.
( ?+ q5 b4 v+ K, L; Rheuristic reasoning.
. o5 s+ S' D6 r1 [) x4 X5 r" i' S, Ea heuristic argument by George Pólya.
7 M# F' R9 R* Y7 N) t: K4 i! j$ {0 |Hilbert’s 23 problems.
! s) H' I3 \+ q) Rhome prime.
+ J' D0 j( m$ d) Nhypothesis H. . ~4 l, W6 F1 f
illegal prime.
 t+ U* L& @- hinconsummate number.
9 C: r0 [3 A7 g, g5 Tinduction.
+ N" T, n% U* Y! ^ J8 jjumping champion.
$ T5 V4 s5 _. X Dk-tuples conjecture, prime. + V, Q3 V; Z7 Z/ V8 i
knots, prime and composite.
# K1 e1 {# O4 D, ~ ~% z) HLandau, Edmund (1877–1938).
+ M- Y1 }+ |* r; l s8 c2 x7 Aleft-truncatable prime.
" o( z6 z: |2 K3 eLegendre, A. M. (1752–1833). ) B" J( q$ J9 F3 q( D b5 k
Lehmer, Derrick Norman (1867–1938). * M; c$ a6 T0 G! s) c1 Z& H# W4 Y
Lehmer, Derrick Henry (1905–1991).
1 {7 }/ Z" E4 b8 rLinnik’s constant. " N7 O6 d) c% i9 c7 w' D0 M
Liouville, Joseph (1809–1882).
- Q/ ^2 i* E" {8 hLittlewood’s theorem. 3 z7 m3 O$ y6 L+ h/ K
the prime numbers race.
) M( q8 n" U5 T' z6 y! dLucas, édouard (1842–1891).
4 w9 z: @4 V3 B+ J& V- Z/ a3 X. F0 c8 Nthe Lucas sequence. 0 v+ J+ q9 a" c
primality testing.
& a8 T# d9 a/ y' g8 MLucas’s game of calculation.
, [/ a3 h- ?* {9 d1 w$ [the Lucas-Lehmer test. 8 p, U! J8 W( w! ~
lucky numbers.
& @# i7 w# L, D9 \8 a/ x& ythe number of lucky numbers and primes. $ H& P- h3 @% ~6 M* B2 ^* G5 D
“random” primes.
! I6 z% B, G4 u, hmagic squares.
0 `6 @2 \: U) z c; tMatijasevic and Hilbert’s 10th problem.
: J3 m$ t! Y o" ?Mersenne numbers and Mersenne primes.
$ R% \, p0 X" D- R- SMersenne numbers.
* Q; `0 h3 e/ v! K7 X7 ohunting for Mersenne primes.
: I+ ^" P/ B9 R; l2 Tthe coming of electronic computers.
+ w# m8 a: i( J3 H( HMersenne prime conjectures. + f' [; Y" n6 D- M" W) q7 U; t. ~
the New Mersenne conjecture.
6 O3 A/ l: K% M0 N* `how many Mersenne primes? . K3 M8 h: w2 S2 J/ G
Eberhart’s conjecture.
! _* e. @- y$ h j. Sfactors of Mersenne numbers.
3 o# D5 o4 v( q4 {2 ^Lucas-Lehmer test for Mersenne primes. & b9 m% E8 N3 u1 D$ [" Z6 ]
Mertens constant.
! n) \6 f+ q- _+ O* gMertens theorem. 1 M2 }" M3 G }5 k7 s
Mills’ theorem. + O: K; U$ o0 n, y$ Z4 w) s
Wright’s theorem. * J1 v. j$ M1 w- @4 Q
mixed bag.
3 A. m% C% P% K# emultiplication, fast. / m i3 @) Z6 r; V2 r% O, L, ^; o. u
Niven numbers.
; G/ E. C: ^ h: e0 u& `* f' [7 [, Nodd numbers as p + 2a2.
* n8 @; H) H6 F W+ d1 b6 `* Z4 UOpperman’s conjecture.
0 S( ~& e0 C* J9 C6 `palindromic primes. " m; x8 O/ x" O( T4 ]
pandigital primes. . Y9 \: O- v( J9 @" K
Pascal’s ** and the binomial coefficients. / U% D$ ]3 ]/ B$ i
Pascal’s ** and Sierpinski’s gasket.
: r- D& u, m/ Z9 bPascal ** curiosities.
6 s1 s! ?7 O: C% [3 i+ E+ b1 Gpatents on prime numbers.
* \, F4 x- h; s& n9 ~- J& {3 @, lPépin’s test for Fermat numbers.
: z& ~$ L* K9 A, ?& e7 i9 J2 { qperfect numbers. ' v/ Q1 e" t0 P. j; Z
odd perfect numbers.
( ~+ q$ w8 A1 A7 Q8 gperfect, multiply.
/ ?1 [% o* X" N) o3 Y2 kpermutable primes. , u% Z5 Q3 C5 c& b. J
π, primes in the decimal expansion of.
1 p& G6 G% N; c0 cPocklington’s theorem.
+ y& [1 a, C" s% z9 dPolignac’s conjectures. 2 W. `; B2 } z! a
Polignac or obstinate numbers. 5 X9 @% t& K! J, ^& {& U% V
powerful numbers. : j7 s5 e2 Q: R; ^+ [1 M) {# L
primality testing. . V- g$ J! t5 T
probabilistic methods.
& K( D2 ~/ l$ `( V" I2 lprime number graph.
0 s4 g& a/ L8 B6 Q ]9 b7 Iprime number theorem and the prime counting function.
0 T" @- ?# O1 Dhistory. * I& w! ^# c5 R$ L# |2 G& }
elementary proof. ' k2 G4 `5 ?8 K8 w$ m) I; j
record calculations. 7 `# B; b# I4 u& `
estimating p(n).
: T1 i/ H9 @' g! z1 s* Q% E6 Q8 @calculating p(n). ' s" A1 L3 _; J) n3 q1 e
a curiosity. # _. m3 D+ c! W/ d' ^& W
prime pretender.
8 y: H1 E0 r Wprimitive prime factor. " g% p7 c8 q3 Q9 v# P: N" Q
primitive roots.
" v+ f& x$ l# O! xArtin’s conjecture. $ w( [( [! {2 c: ~: _; G% u
a curiosity.
( |, M* J* z' a+ p% I. Pprimordial. 6 K' g: |! @) Z" w0 v9 o" _
primorial primes. % }" Y8 \# t# ?1 N! h% Q1 g5 H
Proth’s theorem.
" O6 L% i) B; [( V( K$ J+ mpseudoperfect numbers.
! x6 ]+ @$ ]+ I$ l$ h! D9 zpseudoprimes. - [! @1 z) E. |2 A" s& J8 Q
bases and pseudoprimes. 8 n' c: C. p, p; w. l6 y
pseudoprimes, strong. $ ~" G; n7 S6 Q2 P0 j8 J; C
public key encryption.
+ o' y* P9 d6 v2 W, Npyramid, prime.
; d0 K# Z* f. f+ n- O; ? TPythagorean **s, prime. 5 a q. {1 x6 J- P$ z6 U
quadratic residues. ' E' E0 k! t/ t. c
residual curiosities. ! q2 ]# \; J$ K! L/ F$ R4 b; ^7 l
polynomial congruences.
+ K0 }8 |4 o4 E1 ^; H" _! x+ Vquadratic reciprocity, law of.
$ L, |- p8 U8 {0 h1 O% SEuler’s criterion.
 u% ]+ I% $ z% @+ I, MRamanujan, Srinivasa (1887–1920).
: Z. L& p& m4 `; t0 Fhighly composite numbers. ; p; t" f% m9 N
randomness, of primes. $ ^" B5 a- c9 D8 a# L4 U% h5 w+ b
Von Sternach and a prime random walk.
 X+ |* x+ j4 i @4 V$ U; g, v( ^record primes.
. X9 W7 M' {9 P3 Gsome records.
" c. Z5 o8 {# e3 N& S/ hrepunits, prime.
% T- M1 q* t+ m8 x7 K8 O$ V7 k' kRhonda numbers.
. Q4 ~ ^9 \1 G4 e, X- e! C. k* WRiemann hypothesis. 4 S0 b! G) ^/ m% i) }* |, Y8 l
the Farey sequence and the Riemann hypothesis.
6 W* ^, ]& W% q: q6 N0 O: Z: y- f9 Rthe Riemann hypothesis and σ(n), the sum of divisors function. * L* U0 b0 b3 F# y- ?/ I
squarefree and blue and red numbers.
, $ R. E/ f* \( Dthe Mertens conjecture. + o! v& q9 R+ P/ i0 Y
Riemann hypothesis curiosities.
* [9 c5 R) W: ]) [- IRiesel number. - u3 N/ x6 B% B2 ^$ V; V
right-truncatable prime. % n+ z' i' p, I0 a, {/ F; ?( i
RSA algorithm. ) D& w% g- \ _0 S
Martin Gardner’s challenge.
( u" b R8 h, t( BRSA Factoring Challenge, the New. 0 R6 X" r, b9 y& p: z
Ruth-Aaron numbers.
( K+ H8 _) M2 A5 ~# FScherk’s conjecture.
( }3 U* X8 [/ N" x2 ~" Rsemi-primes. - } n" }7 e6 e d/ k+ D+ a0 L
**y primes.
0 Z3 y+ r6 H( Z1 m, r2 a% JShank’s conjecture. 1 ?; _) N U z" I% f0 ?
Siamese primes.
: B6 H. E5 d! aSierpinski numbers. 0 C, D, L, B, }- A
Sierpinski strings. + j0 V. w( d' P& k( Y
Sierpinski’s quadratic. , S6 l& N: c/ A# Z3 R
Sierpinski’s φ(n) conjecture. 4 c, w; V+ C7 x& Z- y4 D9 Q
Sloane’s On-Line Encyclopedia of Integer Sequences. # ~# ^, M" T! i: F$ D
Smith numbers.
( Y D2 R) q* U: g. fSmith brothers. ) s u- q! Q/ Q. H
smooth numbers.
" z5 w8 x5 R( E$ [7 i" Z5 Y' T7 tSophie Germain primes.
4 M( U. v6 F2 M7 c. E" Bsafe primes.
9 Q& x; K/ T/ p5 ]$ Jsquarefree numbers.
9 x3 K5 i2 j" j: tStern prime.
. r7 K+ I ?, Y( Y/ ` {( }/ {strong law of small numbers.
( K7 P7 r0 n0 S' q0 Wtriangular numbers. 9 V/ x- w" K6 c% Z8 E
trivia.
( x* H: E8 x2 Z3 j0 W y9 Q4 _- Ntwin primes.
7 j8 _# T$ m U1 Q9 r6 n. wtwin curiosities. 0 ^% {4 [: L3 R' w, T4 _, c5 j
Ulam spiral. # Z& k% Y; {8 m5 b; L2 e
unitary divisors. 5 W7 o, ^1 p- N, m1 A4 N0 ^
unitary perfect. % e5 g% h) Y% M$ o- j
untouchable numbers. / n. k( F9 R& }- k6 E
weird numbers.
, z' N, b: V8 J6 L) dWieferich primes.
2 p- e# j {2 s4 m; EWilson’s theorem.
% _3 l. O% Z( W; Htwin primes. ( F. N' }6 m0 O. [$ j s) L1 o
Wilson primes. & a4 ]. r. X# X% f8 F* _
Wolstenholme’s numbers, and theorems. 3 S# q; E$ }* M8 n& g! ~9 U. b
more factors of Wolstenholme numbers.
& w3 ?8 z/ o8 X! dWoodall primes.
3 D' N4 h. _* l. P' c8 Nzeta mysteries: the quantum connection.