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数字的奇妙：素数

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Entries A to Z. ; B: r9 \; e. o* R: D# ^8 a
abc conjecture.
1 n+ _. U* `5 @: P9 \) z& s, J) d, zabundant number. % B" e3 A2 q3 ^7 P- k
AKS algorithm for primality testing.
. r9 j! B/ C2 k: u: m z0 ~aliquot sequences (sociable chains). 1 P6 ^8 J( K+ @
almost-primes. ( b, r' K; ?8 s
amicable numbers. 1 l5 A% k5 B# X4 |2 f7 Y' o( S
amicable curiosities. 1 A3 B. T1 {9 w7 o. [2 X5 G( s
Andrica’s conjecture.
3 v9 a! v( ~1 Xarithmetic progressions, of primes.
$ f. ~. l7 \5 `* j$ R0 C; GAurifeuillian factorization.
- o+ }! ~ {. s% G2 Qaverage prime.
! w i( G' Y, g3 j2 _# ~Bang’s theorem.
: ^- N5 s2 ~% X4 P pBateman’s conjecture.
4 E$ b/ N" ^& C9 O# ^$ _. XBeal’s conjecture, and prize.
 m" h- z7 ]0 P) [5 X. Z/ jBenford’s law.
( f8 q' g& K2 F- p+ zBernoulli numbers.
/ Q# w5 \5 r. C& S# xBernoulli number curiosities. ( `3 p0 J6 H+ Z- W5 h
Bertrand’s postulate. & w8 ^- R% h' N) `" Z% k
Bonse’s inequality.
, [6 K+ x. V0 ^1 Y. a- y/ x! d4 N9 g' DBrier numbers.
# v1 D |) M; B+ TBrocard’s conjecture. 3 B+ [( {1 l+ ^
Brun’s constant. 5 `" G5 \, D: o0 L" K, O5 M
Buss’s function.
( K; s6 X& _, ?8 \% k; ^6 D" w; v QCarmichael numbers.
% y. W' s) A- t R# wCatalan’s conjecture.
6 o1 }" w/ x& t3 d- |Catalan’s Mersenne conjecture. # I- \% F8 q( k
Champernowne’s constant.
/ e" Y9 Q" T7 d% y- n7 I3 Q2 cchampion numbers.
# J+ I; }( o/ L# w5 i5 [Chinese remainder theorem. S' ~$ @0 F6 h# n: }6 {
cicadas and prime periods.
7 q2 _- Z' Y' r1 I* k' acircle, prime. 5 b6 _& e1 {' n
circular prime. 1 B. ^. c1 w1 \! S8 u
Clay prizes, the.
( V! l$ \- C; X1 z5 e( e7 }compositorial.
2 Q4 a3 m7 t ?: Kconcatenation of primes. 0 K6 ^, d4 @$ {: D! e
conjectures. _* ~- D) U. V) I% ^ a
consecutive integer sequence.
 j8 P$ J0 o1 |/ c1 Y7 wconsecutive numbers.
. Q) L9 _6 b2 I5 u: e' `" H" T* mconsecutive primes, sums of. ; ~3 r6 X# }: }$ P
Conway’s prime-producing machine.
3 y* q0 `1 [5 D. f8 acousin primes. * i. |- I# ?2 z* g! e, a
Cullen primes.
) [ P% F" S; c! r7 R/ y5 o UCunningham project.
- q: \7 l/ }, @7 FCunningham chains. 4 C& z) ?$ g( [! Z
decimals, recurring (periodic). 6 L8 ]. d* @+ P l8 f, d
the period of 1/13.
( @- }' V# W; L9 C7 t3 ncyclic numbers. V4 Y/ G( {! V% O' i4 j
Artin’s conjecture. , [9 B/ n. h) H3 _
the repunit connection.
9 ]1 q- u% U2 ^$ V% _magic squares. * {2 L# ?, t( X6 S& l/ \
deficient number. - _" A: i, w: m% E" D; J! a6 t7 w8 m6 ?
deletable and truncatable primes.
- ^; U: F" E4 M# x+ f7 g. `# gDemlo numbers.
% ?' n& `& ~6 H: ndescriptive primes. 0 v3 l" ?& U4 L) [9 N }, G# r
Dickson’s conjecture.
3 @' D. T2 \! \6 wdigit properties.
2 J! S: X8 c7 J! m J+ y' GDiophantus (c. AD 200; d. 284).
& a8 _1 a+ j. q% M: A1 F. CDirichlet’s theorem and primes in arithmetic series. 8 O+ w9 ?8 n: G$ u9 q5 n
primes in polynomials. ! t' ~- d/ {5 c' P
distributed computing.
1 w0 K" i& X7 T) Edivisibility tests.
3 P1 e; f% ? n3 K. ]5 T" mdivisors (factors). 5 y, Z! A7 N, U i
how many divisors? how big is d(n)?
# k: W- u$ B% }* ?8 a+ Y7 `9 x* c& b/ krecord number of divisors.
, f2 ]6 d. F! ucuriosities of d(n). 8 a& `/ s3 b9 i9 F6 N% Q, f
divisors and congruences.
 V* v# }# H' t" rthe sum of divisors function. % o2 b* ]: T" [: i5 \( c/ d
the size of σ(n).
- [( A5 ?- O5 _3 _" ha recursive formula.
6 I& V1 R- x) Q0 R. tdivisors and partitions.
/ P- _2 I' R1 Y5 _& P0 Jcuriosities of σ(n).
1 U1 }9 w: ? {2 [; b7 Eprime factors.
+ @; o7 ?& U+ P% {divisor curiosities. 4 ]; K4 ^' |% ~3 [3 w
economical numbers. * P2 T/ v( \! N, G
Electronic Frontier Foundation.
* k% X( z0 i9 L" |$ welliptic curve primality proving.
3 ]$ G" M+ J. K- w7 o4 Z3 b9 jemirp. : `: m6 u4 s% r! t5 X3 u$ V
Eratosthenes of Cyrene, the sieve of. ( H: W; E( k1 K R6 T
Erd?s, Paul (1913–1996).
# F( w4 x$ E T+ w4 a0 lhis collaborators and Erd?s numbers.
8 J- r/ u- ?* `9 Qerrors. - G/ S- `) P/ k; X! g* A$ I+ v
Euclid (c. 330–270 BC). . I9 o2 _' g" A, ]
unique factorization. & q. _& b. d' ?! V
&Radic;2 is irrational. # S- {) | B% o$ W# `
Euclid and the infinity of primes.
2 d" C; P9 B8 s3 @. Iconsecutive composite numbers. ) d7 w- B D8 x7 k% d
primes of the form 4n +3. % \, _! G Z% ~% H
a recursive sequence. 6 L- l ?5 t+ l% M3 L7 a
Euclid and the first perfect number. $ U$ O s' p# m$ G D0 ?8 ]
Euclidean algorithm.
8 g& Y1 o9 I3 F8 i( IEuler, Leonhard (1707–1783).
- \4 m9 n# g8 S! \8 _4 j( I FEuler’s convenient numbers.
* Z) l& k3 s, r# g5 r4 hthe Basel problem.
 ]9 A3 G8 v3 O0 u1 yEuler’s constant.
 e8 d0 T' _! {# j5 @6 Q5 H$ _Euler and the reciprocals of the primes.
( w) j0 L- Z8 g" Y8 u$ rEuler’s totient (phi) function.
$ m1 |' Y" k# @Carmichael’s totient function conjecture.
3 E4 h O9 M1 H. Y& D' p3 X; zcuriosities of φ(n).
& i$ ?/ X: w$ _3 i$ _" P$ P) l1 F. jEuler’s quadratic. : F, ]5 ]* K$ N" E/ O8 t
the Lucky Numbers of Euler. ' o* m3 _8 ?8 k, w" p0 O" P
factorial. ) E \, ~: X/ w
factors of factorials. % E" v4 g; ~) [/ N) X( X! u2 ?
factorial primes. * G# q/ p; }6 N/ u
factorial sums.
0 A" R) m' N2 r P3 W# Cfactorials, double, triple . . . .
" n5 f" w- V5 N) ~factorization, methods of. : I2 J) G; i5 S: b5 @$ t6 a- f
factors of particular forms. * }2 v. T$ N7 @ \) ?$ y- q
Fermat’s algorithm. ) n. J' E3 ^; Q# _, b) X7 H4 b* D$ O
Legendre’s method.
4 r: j4 h! _8 ^3 q) L3 wcongruences and factorization.
0 l) g: K9 d/ c) g- C' f4 W# Fhow difficult is it to factor large numbers? A/ k* F: e6 w$ E% ^* W
quantum computation. * a5 L' j9 e; H/ W) y/ H
Feit-Thompson conjecture. * j) {% T" g6 P, {
Fermat, Pierre de (1607–1665). 3 T0 _3 N/ W' |, c
Fermat’s Little Theorem. ) L6 o) o' g/ z$ U8 Q
Fermat quotient.
# x$ I2 A% N! x( vFermat and primes of the form x2 + y2. * B; v$ C, e0 W) ~% }2 o1 O
Fermat’s conjecture, Fermat numbers, and Fermat primes.
1 `% y6 q5 @- J6 e! nFermat factorization, from F5 to F30. - i% j5 d( b7 P. ]- }; [7 c
Generalized Fermat numbers. * W! w; V/ ]( i# E* p" e6 u
Fermat’s Last Theorem. , I( M: _: n* s5 [' ?
the first case of Fermat’s Last Theorem. & T( ]& r# I, @" ?1 T$ n
Wall-Sun-Sun primes. L* B0 j+ g! J5 h' G8 I" e
Fermat-Catalan equation and conjecture.
* U4 j9 |8 _" t( M, k* KFibonacci numbers.
( f @: X) Y1 I+ G/ b/ gdivisibility properties. ; ?: v! b" d [" Z& n
Fibonacci curiosities. ' B. v5 m. [) O- J' x7 s( ?
édouard Lucas and the Fibonacci numbers.
9 u8 r' ~6 f2 GFibonacci composite sequences.
8 l% Y- i. x' B" Zformulae for primes.
8 q" R2 c/ j4 WFortunate numbers and Fortune’s conjecture.
5 Z) |+ N3 g/ ?9 H' W, z) M" ^) tgaps between primes and composite runs. & k+ j! u2 b3 K% f
Gauss, Johann Carl Friedrich (1777–1855).
1 u$ y7 j* w( ]+ B) |Gauss and the distribution of primes.
" W3 h: j! {/ LGaussian primes. 4 G4 x; G2 A6 R6 y0 h P/ W
Gauss’s circle problem.
' O2 A+ y# {5 B# EGilbreath’s conjecture. 7 A9 S# u0 y* ]; i: M6 V
GIMPS—Great Internet Mersenne Prime Search. * k( n7 M/ S; N/ ~
Giuga’s conjecture.
1 p4 H6 v' l/ c5 b: lGiuga numbers.
4 {9 K* m) @% x! l" s- lGoldbach’s conjecture. , i5 H1 Z6 u' [, P
good primes.
2 T# p/ Q4 p" ?7 ]+ YGrimm’s problem.
0 H) F, m; k7 [6 K- W- {, D$ D" e& ]Hardy, G. H. (1877–1947).
, N/ G$ r! C1 MHardy-Littlewood conjectures. , C* `( `2 c/ ?- X/ S, e5 r% g
heuristic reasoning. ) |$ t9 Y$ E$ u' g6 r8 ^
a heuristic argument by George Pólya. - E$ q! t6 D; l4 o& E& I
Hilbert’s 23 problems.
0 |. h/ E; p) fhome prime. 4 M; I! Q6 a; ]2 G' e5 y
hypothesis H.
; c. D6 }7 i$ b' N% _3 S# r4 m$ billegal prime. ( d0 k* n/ M" Q
inconsummate number. ' b; \% S' J$ |0 R; g/ `
induction.
# u4 D2 a2 t# U& ejumping champion.
: H! @- r5 ]& H/ x- qk-tuples conjecture, prime. 7 n1 m# y: X$ c! r7 t% m
knots, prime and composite. % w* D/ Z4 l3 S9 [, |1 k, |
Landau, Edmund (1877–1938). 0 C! @& p' V; ?) w9 V! j
left-truncatable prime.
( M0 Q1 l% y; q" Z; ~. SLegendre, A. M. (1752–1833).
8 p+ h0 Q0 Q( ?# {; zLehmer, Derrick Norman (1867–1938).
* P A0 `% W+ W0 v7 u! LLehmer, Derrick Henry (1905–1991). 7 N. D6 h" S' b P9 @. T0 }- h
Linnik’s constant. 6 u$ v/ G+ {8 F5 ]- F
Liouville, Joseph (1809–1882).
' e4 U4 H! P, m7 e! n. O1 yLittlewood’s theorem.
) c% \) T6 P: T7 ^the prime numbers race. 9 S1 q! m" ~4 {# C2 f# \. r
Lucas, édouard (1842–1891). $ Z5 E7 A/ m/ t+ n- E+ X+ r0 }
the Lucas sequence. * G9 S& r; n+ I2 C) S$ g6 a
primality testing. ) {: B: o% \0 Y9 h/ d
Lucas’s game of calculation.
1 r7 M% z& F; F8 t; ^the Lucas-Lehmer test.
5 p+ `' v. j) b7 olucky numbers.
2 E' g5 B% i" nthe number of lucky numbers and primes. ) q5 K# ~% K% N0 r$ I) d7 X! t K
“random” primes.
- L3 @* s6 x) e' hmagic squares. ! r5 c- z. m- _% N
Matijasevic and Hilbert’s 10th problem.
( ~2 J1 x1 C, ?" f( ^+ HMersenne numbers and Mersenne primes.
2 _) {0 Z# s: p7 J s+ lMersenne numbers.
6 |: g8 n" M; m! u9 [hunting for Mersenne primes.
+ d% U F j0 l* Q* ~$ u( Q/ rthe coming of electronic computers.
9 g" a8 j7 d0 t4 e, k3 n7 IMersenne prime conjectures.
- y! `' b( G% j, ^the New Mersenne conjecture. 2 q5 t, P0 n; T7 E4 Y8 ?2 Q: `
how many Mersenne primes?
& c4 [+ H: _0 V* L, @/ v7 T! z( WEberhart’s conjecture.
/ H, Z0 a" G5 c5 @3 Y7 `factors of Mersenne numbers.
" w4 f3 Z; \/ W$ U/ I. W5 n% dLucas-Lehmer test for Mersenne primes.
" E# C" T, c, t2 eMertens constant.
6 y* F6 S/ ^+ ]- |) k1 xMertens theorem. 3 o; u: O% `8 F: S3 Z; g6 a) U
Mills’ theorem.
+ L( k1 X: ^. @; f8 ZWright’s theorem. ! @( }. |8 ]0 h, U r% Y0 ]! R4 [% `
mixed bag.
2 V; a5 B q& _! [; C( mmultiplication, fast. / k2 \9 \& p$ j0 B9 [
Niven numbers. & M) N# o+ y/ b0 Z, R7 ?) l
odd numbers as p + 2a2.
6 h1 m4 w) P* c. V3 qOpperman’s conjecture.
7 c# b' g' c! F' F6 y" R0 L( Ypalindromic primes. 0 D7 c) V U" v5 m( P
pandigital primes. 0 n; S) E& U$ s3 i% L5 j
Pascal’s ** and the binomial coefficients.
6 a+ G `1 A7 Q' g4 V9 zPascal’s ** and Sierpinski’s gasket.
& x; |. l( j2 w9 \( U( y- J* ?Pascal ** curiosities.
/ q% H3 o8 J, X; Qpatents on prime numbers.
. r" s, T z8 ~Pépin’s test for Fermat numbers.
) ?# c/ ?- T$ j2 _$ o2 T; a$ T) Tperfect numbers. ( s9 L$ H2 w& k2 u
odd perfect numbers.
2 i3 h0 h9 P; F% ~+ Q. {0 y) ~perfect, multiply. * P/ b; J! M/ I! d. q( o5 `3 T+ ~# c
permutable primes. # n0 t. w; z" _
π, primes in the decimal expansion of. 7 M5 l& V7 Y/ b5 j6 k, Z
Pocklington’s theorem.
0 Z' w5 `' K) V: `Polignac’s conjectures.
+ q( ~6 \' W& x- l; W' F0 ]$ u: xPolignac or obstinate numbers. ; P7 c% {' |2 F
powerful numbers. : W2 E4 N: ^9 E& y+ u) m/ d% [
primality testing.
) ]; d: i, Y& o3 ^probabilistic methods.
4 T9 T6 o# {* A; w: b; dprime number graph. ' a" Q( s* D5 n# p; m6 L0 o0 _
prime number theorem and the prime counting function. , l& f4 \. I) R/ a5 c h
history.
1 o l; ~, A0 x& Ielementary proof.
. y: A8 b3 B1 B$ k, T& R7 Rrecord calculations. , d. J4 V: z* D0 l
estimating p(n).
1 z/ y5 C, K* v7 \8 `7 d4 \calculating p(n). 2 }: T. N. L: U4 P# Z5 I7 p1 K& n/ T$ m
a curiosity. 9 S3 T( j7 ^$ {
prime pretender.
; }2 c) }3 n7 z- iprimitive prime factor.
9 C( v# V% y1 N' oprimitive roots. {5 ^/ V; J$ x, n7 W& |
Artin’s conjecture. 5 _+ |, ]6 I D! x: A/ Y
a curiosity.
& e2 g% _3 g2 B3 \3 T. H- Iprimordial. + S6 b- L# U5 }
primorial primes.
. Z2 {) \. a6 g S' R8 pProth’s theorem.
6 E" K' G4 |1 n4 v& `pseudoperfect numbers.
2 O) i7 {" j& Ypseudoprimes. 1 n Q5 q( r. C$ t4 ]
bases and pseudoprimes.
; `+ f7 u$ N5 @4 v( Qpseudoprimes, strong. : ]7 V' e* [; `; k7 L7 I6 H
public key encryption. " t' D1 z4 o; c$ d
pyramid, prime. 0 N7 G a+ F' H: X0 j& X
Pythagorean **s, prime. 0 H. {+ f' z( y
quadratic residues. 4 H; P& Q& d2 Q6 s& v
residual curiosities. ; _( H/ W0 _+ O& P* V
polynomial congruences.
' b. a2 N3 ~& q4 @quadratic reciprocity, law of. * o! N% k0 A; n/ ^
Euler’s criterion. 6 C* C# t& y$ d+ n$ K9 X/ [
Ramanujan, Srinivasa (1887–1920). ; _2 N: f1 {0 s3 ~7 ?
highly composite numbers. U& S+ Q+ o- ]' c0 t8 t- \
randomness, of primes. ; X+ D: b$ } r' E+ p
Von Sternach and a prime random walk.
+ H3 f, t, K% ?% T2 Jrecord primes. ' [, R- ]- A. |
some records. & ? g6 W+ o% _/ _. D5 h2 K6 s
repunits, prime.
& D ]* k0 C, n! P* gRhonda numbers. + d( U! @" G! Q6 l
Riemann hypothesis. 0 ]+ {/ k+ B3 T5 t2 o4 n: O$ z2 ?
the Farey sequence and the Riemann hypothesis. 5 @; g5 ? X; D: }# E+ u( Y
the Riemann hypothesis and σ(n), the sum of divisors function.
( M/ S$ ]; y2 ^1 Z% h1 u B- L9 hsquarefree and blue and red numbers.
7 @6 q0 p- y/ {- R U) q$ M" Gthe Mertens conjecture. . M) c. \0 p" ^) ^6 w
Riemann hypothesis curiosities. ; y+ v' g& G/ ?4 C8 u
Riesel number.
* w9 O) B r g, f& e8 h& _right-truncatable prime.
/ `3 U& I6 V8 f4 w9 L; ~0 n2 z/ ?RSA algorithm. / Q) B* I0 w# b8 J
Martin Gardner’s challenge.
! ? b: i8 |5 j- I( y( _0 fRSA Factoring Challenge, the New. ^3 j6 A8 X5 C6 f4 R
Ruth-Aaron numbers.
0 M; A, Y9 n( A8 O1 yScherk’s conjecture. : z H5 B$ B! S' I4 _
semi-primes. 6 M. h. g) ` E7 A
**y primes.
. w/ h) z& s' ` V+ NShank’s conjecture.
$ f3 \; n! J8 j5 CSiamese primes. 1 V1 Q1 |/ t/ S, b1 E
Sierpinski numbers.
5 E2 e. u- \% B- j9 gSierpinski strings. . J) [) J% I4 i7 Y
Sierpinski’s quadratic. : |* t O* c+ ^* [* Q N ?
Sierpinski’s φ(n) conjecture.
 y0 Y7 F4 u/ a. }Sloane’s On-Line Encyclopedia of Integer Sequences.
9 d& d! n. R3 C7 V! J2 wSmith numbers. 1 u# f$ v. a1 n5 I$ P( k
Smith brothers.
2 |# W& h) K1 N' X3 ssmooth numbers. ) w9 b+ e8 \" h
Sophie Germain primes.
' Q( ]5 o& L2 @1 U0 jsafe primes.
' S# D. c/ O* s6 a& o: J: |squarefree numbers. : N N1 c" T1 \. ]) [
Stern prime.
9 _$ k) r2 W; j7 istrong law of small numbers.
( `4 q% q/ }: Y' J/ etriangular numbers.
& L, h, D/ u; Y: \trivia.
* p! c6 I& i Htwin primes.
$ a1 G/ B/ q" Y7 Y! n8 O/ ttwin curiosities. 6 G& D/ D7 ^5 ~+ ~
Ulam spiral.
; b, a2 j/ P% @4 X/ m- Kunitary divisors.
. K1 O8 x: \" }; c& n/ o0 hunitary perfect. 1 c. Z6 _; @9 F' Y
untouchable numbers. j! w# ^) Z0 S% \, @
weird numbers.
3 m9 E m( q0 ]! b% @$ {8 |0 V1 RWieferich primes.
; q& I; h; k) ^( \% n4 p. IWilson’s theorem. # ^! O6 {2 O, h9 b1 H" y
twin primes. 7 c' [ V* A7 l" r j4 n8 j
Wilson primes.
& [& G% z. j0 i6 A# iWolstenholme’s numbers, and theorems. * m; h9 y0 l, ]7 ]
more factors of Wolstenholme numbers. + R# I/ c( ^0 S( _- A, ]& X- ~
Woodall primes. ; r* }7 u6 X8 f. D& e. E, J
zeta mysteries: the quantum connection.