2000 (number)
(Redirected from 2,024)
| ||||
|---|---|---|---|---|
| Cardinal | two thousand | |||
| Ordinal | 2000th (two thousandth) | |||
| Factorization | 24 × 53 | |||
| Greek numeral | ,Β´ | |||
| Roman numeral | MM, mm | |||
| Unicode symbol(s) | MM, mm | |||
| Binary | 111110100002 | |||
| Ternary | 22020023 | |||
| Senary | 131326 | |||
| Octal | 37208 | |||
| Duodecimal | 11A812 | |||
| Hexadecimal | 7D016 | |||
| Armenian | Ս | |||
| Egyptian hieroglyph | 𓆽 | |||
Look up two thousand in Wiktionary, the free dictionary.
2000 (two thousand) is a natural number following 1999 and preceding 2001.
In mathematics
[edit]2000 is:
- the highest number expressible using only two unmodified characters in Roman numerals (MM)
- an Achilles number[1]
- smallest four digit eban number[2]
- the sum of all the nban numbers (sequence A008537 in the OEIS)[3]
Selected numbers in the range 2001–2999
[edit]2001 to 2099
[edit]2002
[edit]2002 = 2 × 7 × 11 × 13. It is a palindromic number in decimal, base 76, 90, 142, and 11 other non-trivial bases.
2003
[edit]2003 is a Sophie Germain prime.
2005
[edit]2005 = 5 × 401. It is a vertically symmetric number.
2011
[edit]2011 is a sexy prime with 2017. It is also the sum of eleven consecutive primes: 2011 = 157 + 163 + 167 + 173 + 179 + 181 + 191 + 193 + 197 + 199 + 211
2015
[edit]2015 = 5 × 13 × 31. It is a Lucas–Carmichael number.[4]
2016
[edit]2016 = 25 × 32 × 7. It is the second-smallest Erdős–Nicolas number[5]and a triangular number.[6]
2017
[edit]2017 is a sexy prime with 2011.
2024
[edit]2024 = 23 × 11 × 23. It is a tetrahedral number.[7]
2025
[edit]2025 = 452, square of the sum of the first nine positive integers (and therefore sum of the cubes of the first nine positive integers, by Nicomachus's theorem), centered octagonal number,[8] lowest number with exactly 15 odd divisors.[9] Sum of odd numbers from 1 to 89.
2027
[edit]2027 is a super-prime and a safe prime.[10]
2029
[edit]2029 is a prime number and a member of the Mian–Chowla sequence.[11]
2031
[edit]2031 = 3 × 677. It is a centered pentagonal number.[12]
2035
[edit]2035 = 5 × 11 × 37. It is a Wolstenholme number.[13]
2036
[edit]2036 = 22 × 509. It is an Eulerian number.[14]
2039
[edit]2039 is a Sophie Germain prime and a safe prime.[10]
2047
[edit]2047 = 23 × 89 = 211 − 1 and it is the first Mersenne number that is composite for a prime exponent. It is a super-Poulet number,[15] a Woodall number,[16] a decagonal number,[17] and a centered octahedral number.[18]
2053
[edit]2053 is a prime number and a star number.
2056
[edit]2056 = 23 × 257. It is the magic constant of an n × n normal magic square and the n-queens problem for n = 16.
2063
[edit]2063 is a Sophie Germain prime, a safe prime,[10] and a super-prime.
2066
[edit]2066 = 2 × 1033. It is a Bell number.[19]
2069
[edit]2069 is a Sophie Germain prime.
2080
[edit]2080 = 25 × 5 × 13. It is a triangular number.
2081, 2083, 2087, and 2089
[edit]2081, 2083, 2087, and 2089 form a prime quadruplet.
2081 is also a super-prime.
2099
[edit]2099 is a super-prime, a safe prime,[10] and a highly cototient number.[20]
2100 to 2199
[edit]2101
[edit]2101 = 11 × 191. It is a centered heptagonal number.[21]
2107 and 2108
[edit]2107 and 2108 form a Ruth–Aaron pair under the first definition.
2109
[edit]2109 = 3 × 19 × 37. It is a square pyramidal number.[22]
2113
[edit]2113 is a Proth prime[23] and a centered square number.[24]
2125
[edit]2125 = 53 × 17. It is a nonagonal number.[25]
2129
[edit]2129 is a Sophie Germain prime.
2141
[edit]2141 is a Sophie Germain prime.
2145
[edit]2145 = 3 × 5 × 11 × 13. It is a triangular number.
2160
[edit]2160 = 24 × 33 × 5. It is a largely composite number.[26]
2176
[edit]2176 = 27 × 17. It is a pentagonal pyramidal number[27] and a centered pentagonal number.[12]
2179
[edit]2179 is a prime number and a Wedderburn–Etherington number.[28]
2187
[edit]2187 = 37. It is a vampire number[29] and a perfect totient number.[30]
2188
[edit]2188 = 22 × 547. It is a Motzkin number.[31]
2199
[edit]2199 = 3 × 733. It is a perfect totient number.[30]
2200 to 2299
[edit]2201
[edit]2201 – only known non-palindromic number whose cube is palindromic; also no known fourth or higher powers are palindromic for non-palindromic numbers
2207
[edit]2207 – safe prime,[10] Lucas prime[32]
2208
[edit]2208 – Keith number[33]
2209
[edit]2209 = 472, palindromic in base 14 (B3B14), centered octagonal number[8]
2211
[edit]2211 – triangular number
2221
[edit]2221 – super-prime, happy number
2223
[edit]2223 – Kaprekar number[34]
2232
[edit]2232 – decagonal number[17]
2245
[edit]2245 – centered square number[24]
2254
[edit]2254 – member of the Mian–Chowla sequence[11]
2255
[edit]2255 – octahedral number[35]
2269
[edit]2269 – super-prime, cuban prime[36]
2273
[edit]2273 – Sophie Germain prime
2276
[edit]2276 _ centered heptagonal number[21]
2278
[edit]2278 – triangular number
2281
[edit]2281 – star number,
2287
[edit]2287 – balanced prime[37]
2299
[edit]2299 – member of a Ruth–Aaron pair with 2300 (first definition)
2300 to 2399
[edit]2300
[edit]2300 – tetrahedral number,[7] member of a Ruth–Aaron pair with 2299 (first definition)
2301
[edit]2301 – nonagonal number[25]
2309
[edit]2309 – primorial prime, twin prime with 2311, Mertens function zero, highly cototient number[20]
2310
[edit]2311
[edit]2311 – primorial prime, twin prime with 2309
2326
[edit]2326 – centered pentagonal number[12]
2328
[edit]2328 – sum of the totient function for the first 87 integers, the number of groups of order 128[39]
2331
[edit]2331 – centered cube number[40]
2339
[edit]2339 – Sophie Germain prime, twin prime with 2341
2341
[edit]2341 – super-prime, twin prime with 2339
2346
[edit]2346 – triangular number
2351
[edit]2351 – Sophie Germain prime, super-prime
2352
[edit]2352 – pronic number[41]
2357
[edit]2357 – Smarandache–Wellin prime[42]
2378
[edit]2378 – Pell number[43]
2379
[edit]2379 – member of the Mian–Chowla sequence[11]
2381
[edit]2381 – super-prime, centered square number[24]
2393
[edit]2393 – Sophie Germain prime
2399
[edit]2399 – Sophie Germain prime
2400 to 2499
[edit]- 2401 = 492 = 74, centered octagonal number[8]
- 2415 – triangular number
- 2417 – super-prime, balanced prime[37]
- 2425 – decagonal number[17]
- 2437 – cuban prime,[36] largest right-truncatable prime in base 5
- 2447 – safe prime[10]
- 2450 – pronic number[41]
- 2458 – centered heptagonal number[21]
- 2459 – Sophie Germain prime, safe prime[10]
- 2465 – magic constant of n × n normal magic square and n-queens problem for n = 17, Carmichael number[44]
- 2470 – square pyramidal number[22]
- 2477 – super-prime, cousin prime
- 2481 – centered pentagonal number[12]
- 2484 – nonagonal number[25]
- 2485 – triangular number, number of planar partitions of 13[45]
- 2491 = member of Ruth–Aaron pair with 2492 under second definition
- 2492 – member of Ruth–Aaron pair with 2491 under second definition
2500 to 2599
[edit]- 2510 – member of the Mian–Chowla sequence[11]
- 2513 – member of the Padovan sequence[46]
- 2520 – superior highly composite number; smallest number divisible by numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, and 12; colossally abundant number; Harshad number in several bases. It is also the highest number with more divisors than any number less than double itself (sequence A072938 in the OEIS). Not only it is the 7th (and last) number with more divisors than any number double itself but is also the 7th number that is highly composite and the lowest common multiple of a consecutive set of integers from 1 (sequence A095921 in the OEIS) which is a property the previous number with this pattern of divisors does not have (360). That is, although 360 and 2520 both have more divisors than any number twice themselves, 2520 is the lowest number divisible by both 1 to 9 and 1 to 10, whereas 360 is not the lowest number divisible by 1 to 6 (which 60 is) and is not divisible by 1 to 7 (which 420 is). It is also the 6th and largest highly composite number that is a divisor of every higher highly composite number (sequence A106037 in the OEIS).
- 2521 – star prime, centered square number[24]
- 2530 – Leyland number[47]
- 2543 – Sophie Germain prime, sexy prime with 2549
- 2549 – Sophie Germain prime, super-prime, sexy prime with 2543
- 2550 – pronic number[41]
- 2556 – triangular number
- 2579 – safe prime,[10] palindromic prime in bases 5, 25, and 28: 2579 = 403045 = 43425 = 38328
- 2580 – Keith number,[33]
- 2584 – Fibonacci number,[48]
2600 to 2699
[edit]- 2600 – tetrahedral number,[7] member of a Ruth–Aaron pair with 2601 (first definition)
- 2601 = 512, member of a Ruth–Aaron pair with 2600 (first definition)
- 2609 – super-prime
- 2620 – telephone number, amicable number with 2924
- 2625 = a centered octahedral number[18]
- 2626 – decagonal number[17]
- 2628 – triangular number
- 2641 – centered pentagonal number[12]
- 2647 – super-prime, centered heptagonal number[21]
- 2652 – pronic number[41]
- 2665 – centered square number[24]
- 2674 – nonagonal number[25]
- 2677 – balanced prime[37]
- 2680 – number of 11-queens problem solutions
- 2683 – super-prime
- 2689 – Proth prime[23]
- 2693 – Sophie Germain prime
- 2699 – Sophie Germain prime
2700 to 2799
[edit]- 2701 – triangular number, super-Poulet number[15]
- 2702 – sum of the totient function for the first 94 integers
- 2707 – strong prime
- 2719 – super-prime, largest known odd number which cannot be expressed in the form x2 + y2 + 10z2 where x, y and z are integers.[49] In 1997 it was conjectured that this is also the largest such odd number.[50] It is now[when?] known this is true if the generalized Riemann hypothesis is true.[51]
- 2728 – Kaprekar number[34]
- 2729 – highly cototient number[20]
- 2731 – the only Wagstaff prime with four digits,[52] Jacobsthal prime
- 2736 – octahedral number[35]
- 2741 – Sophie Germain prime, 400th prime number
- 2744 = 143, palindromic in base 13 (133113)
- 2747 – sum of the first 38 primes
- 2749 – super-prime, cousin prime with 2753
- 2753 – Sophie Germain prime, Proth prime[23]
- 2756 – pronic number[41]
- 2774 – sum of the totient function for the first 95 integers
- 2775 – triangular number
- 2780 – member of the Mian–Chowla sequence[11]
- 2783 – member of a Ruth–Aaron pair with 2784 (first definition)
- 2784 – member of a Ruth–Aaron pair with 2783 (first definition)
- 2791 – cuban prime[36]
2800 to 2899
[edit]- 2801 – first base 7 repunit prime
- 2803 – super-prime
- 2806 – centered pentagonal number,[12] sum of the totient function for the first 96 integers
- 2809 = 532, centered octagonal number[8]
- 2813 – centered square number[24]
- 2819 – Sophie Germain prime, safe prime, sum of seven consecutive primes (383 + 389 + 397 + 401 + 409 + 419 + 421)[10]
- 2821 – Carmichael number[44]
- 2835 – decagonal number[17]
- 2843 – centered heptagonal prime[53]
- 2850 – triangular number
- 2862 – pronic number[41]
- 2870 – square pyramidal number[22]
- 2871 – nonagonal number[25]
- 2872 – tetranacci number[54]
- 2875 – number of lines on a quintic threefold[55]
- 2879 – safe prime[10]
- 2897 – super-prime, Markov prime[56]
2900 to 2999
[edit]- 2903 – Sophie Germain prime, safe prime,[10] balanced prime[37]
- 2909 – super-prime
- 2915 – Lucas–Carmichael number[4]
- 2924 – amicable number with 2620
- 2925 – magic constant of n × n normal magic square and n-queens problem for n = 18, tetrahedral number,[7] member of the Mian-Chowla sequence[11]
- 2926 – triangular number
- 2939 – Sophie Germain prime
- 2963 – Sophie Germain prime, safe prime, balanced prime[37]
- 2965 – greater of second pair of Smith brothers, centered square number[24]
- 2969 – Sophie Germain prime
- 2970 – harmonic divisor number,[57] pronic number[41]
- 2976 – centered pentagonal number[12]
- 2999 – safe prime
Prime numbers
[edit]There are 127 prime numbers between 2000 and 3000:[58][59]
- 2003, 2011, 2017, 2027, 2029, 2039, 2053, 2063, 2069, 2081, 2083, 2087, 2089, 2099, 2111, 2113, 2129, 2131, 2137, 2141, 2143, 2153, 2161, 2179, 2203, 2207, 2213, 2221, 2237, 2239, 2243, 2251, 2267, 2269, 2273, 2281, 2287, 2293, 2297, 2309, 2311, 2333, 2339, 2341, 2347, 2351, 2357, 2371, 2377, 2381, 2383, 2389, 2393, 2399, 2411, 2417, 2423, 2437, 2441, 2447, 2459, 2467, 2473, 2477, 2503, 2521, 2531, 2539, 2543, 2549, 2551, 2557, 2579, 2591, 2593, 2609, 2617, 2621, 2633, 2647, 2657, 2659, 2663, 2671, 2677, 2683, 2687, 2689, 2693, 2699, 2707, 2711, 2713, 2719, 2729, 2731, 2741, 2749, 2753, 2767, 2777, 2789, 2791, 2797, 2801, 2803, 2819, 2833, 2837, 2843, 2851, 2857, 2861, 2879, 2887, 2897, 2903, 2909, 2917, 2927, 2939, 2953, 2957, 2963, 2969, 2971, 2999
References
[edit]- ↑ Sloane, N. J. A. (ed.). "Sequence A052486 (Achilles numbers - powerful but imperfect: if n = Product(p_i^e_i) then all e_i > 1 (i.e., powerful), but the highest common factor of the e_i is 1, i.e., not a perfect power)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ↑ Sloane, N. J. A. (ed.). "Sequence A006933 ('Eban' numbers (the letter 'e' is banned!))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ↑ Sloane, N. J. A. (ed.). "Sequence A008537 (Numbers that do not contain the letter 'n'))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- 1 2 Sloane, N. J. A. (ed.). "Sequence A006972 (Lucas-Carmichael numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ↑ Sloane, N. J. A. (ed.). "Sequence A194472 (Erdős-Nicolas numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ↑ Sloane, N. J. A. (ed.). "Sequence A000217 (Triangular numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- 1 2 3 4 Sloane, N. J. A. (ed.). "Sequence A000292 (Tetrahedral numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- 1 2 3 4 Sloane, N. J. A. (ed.). "Sequence A016754 (Odd squares: a(n) = (2n+1)^2. Also centered octagonal numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ↑ Sloane, N. J. A. (ed.). "Sequence A038547 (Least number with exactly n odd divisors.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- 1 2 3 4 5 6 7 8 9 10 11 Sloane, N. J. A. (ed.). "Sequence A005385 (Safe primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- 1 2 3 4 5 6 Sloane, N. J. A. (ed.). "Sequence A005282 (Mian-Chowla sequence)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- 1 2 3 4 5 6 7 Sloane, N. J. A. (ed.). "Sequence A005891 (Centered pentagonal numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ↑ Sloane, N. J. A. (ed.). "Sequence A007408 (Wolstenholme numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ↑ Sloane, N. J. A. (ed.). "Sequence A000295 (Eulerian numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- 1 2 Sloane, N. J. A. (ed.). "Sequence A050217 (Super-Poulet numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ↑ Sloane, N. J. A. (ed.). "Sequence A003261 (Woodall numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- 1 2 3 4 5 Sloane, N. J. A. (ed.). "Sequence A001107 (10-gonal (or decagonal) numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- 1 2 Sloane, N. J. A. (ed.). "Sequence A001845 (Centered octahedral numbers (crystal ball sequence for cubic lattice))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ↑ Sloane, N. J. A. (ed.). "Sequence A011971 (Aitken's array)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- 1 2 3 Sloane, N. J. A. (ed.). "Sequence A100827 (Highly cototient numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- 1 2 3 4 Sloane, N. J. A. (ed.). "Sequence A069099 (Centered heptagonal numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-13.
- 1 2 3 Sloane, N. J. A. (ed.). "Sequence A000330 (Square pyramidal numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-13.
- 1 2 3 Sloane, N. J. A. (ed.). "Sequence A080076 (Proth primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-13.
- 1 2 3 4 5 6 7 Sloane, N. J. A. (ed.). "Sequence A001844 (Centered square numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-13.
- 1 2 3 4 5 Sloane, N. J. A. (ed.). "Sequence A001106 (9-gonal (or enneagonal or nonagonal) numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-13.
- ↑ Sloane, N. J. A. (ed.). "Sequence A067128 (Ramanujan's largely composite numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ↑ Sloane, N. J. A. (ed.). "Sequence A002411 (Pentagonal pyramidal numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-13.
- ↑ Sloane, N. J. A. (ed.). "Sequence A001190 (Wedderburn-Etherington numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-13.
- ↑ Sloane, N. J. A. (ed.). "Sequence A014575 (Vampire numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-13.
- 1 2 Sloane, N. J. A. (ed.). "Sequence A082897 (Perfect totient numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-13.
- ↑ Sloane, N. J. A. (ed.). "Sequence A001006 (Motzkin numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-13.
- ↑ Sloane, N. J. A. (ed.). "Sequence A005479 (Prime Lucas numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-13.
- 1 2 Sloane, N. J. A. (ed.). "Sequence A007629 (Repfigit (REPetitive FIbonacci-like diGIT) numbers (or Keith numbers))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-13.
- 1 2 Sloane, N. J. A. (ed.). "Sequence A006886 (Kaprekar numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-13.
- 1 2 Sloane, N. J. A. (ed.). "Sequence A005900 (Octahedral numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-13.
- 1 2 3 Sloane, N. J. A. (ed.). "Sequence A002407 (Cuban primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-13.
- 1 2 3 4 5 Sloane, N. J. A. (ed.). "Sequence A006562 (Balanced primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-13.
- ↑ Sloane, N. J. A. (ed.). "Sequence A002110 (Primorial numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-13.
- ↑ "The Small Groups library". Archived from the original on 2007-02-04. Retrieved 2008-01-22..
- ↑ Sloane, N. J. A. (ed.). "Sequence A005898 (Centered cube numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-13.
- 1 2 3 4 5 6 7 Sloane, N. J. A. (ed.). "Sequence A002378 (Oblong (or promic, pronic, or heteromecic) numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ↑ Sloane, N. J. A. (ed.). "Sequence A069151 (Concatenations of consecutive primes, starting with 2, that are also prime)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-13.
- ↑ Sloane, N. J. A. (ed.). "Sequence A000129 (Pell numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-13.
- 1 2 Sloane, N. J. A. (ed.). "Sequence A002997 (Carmichael numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-13.
- ↑ Sloane, N. J. A. (ed.). "Sequence A000219 (Number of planar partitions (or plane partitions) of n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ↑ Sloane, N. J. A. (ed.). "Sequence A000931 (Padovan sequence)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-13.
- ↑ Sloane, N. J. A. (ed.). "Sequence A076980 (Leyland numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-13.
- ↑ Sloane, N. J. A. (ed.). "Sequence A000045 (Fibonacci numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-13.
- ↑ "Odd numbers that are not of the form x^2+y^2+10*z^2.". The Online Encyclopedia of Integer Sequences. The OEIS Foundation, Inc. Retrieved 13 November 2012.
- ↑ Ono, Ken (1997). "Ramanujan, taxicabs, birthdates, zipcodes and twists" (PDF). American Mathematical Monthly. 104 (10): 912–917. doi:10.2307/2974471. JSTOR 2974471. Archived from the original (PDF) on 15 October 2015. Retrieved 11 November 2012.
- ↑ Ono, Ken; K Soundararajan (1997). "Ramanujan's ternary quadratic forms" (PDF). Inventiones Mathematicae. 130 (3): 415–454. Bibcode:1997InMat.130..415O. doi:10.1007/s002220050191. S2CID 122314044. Archived from the original (PDF) on 18 July 2019. Retrieved 12 November 2012.
- ↑ Sloane, N. J. A. (ed.). "Sequence A000979 (Wagstaff primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-13.
- ↑ Sloane, N. J. A. (ed.). "Sequence A144974 (Centered heptagonal prime numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-13.
- ↑ Sloane, N. J. A. (ed.). "Sequence A000078 (Tetranacci numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-13.
- ↑ Pandharipande, Rahul (1998), "Rational curves on hypersurfaces (after A. Givental)", Astérisque, 1997/98 (252): 307–340, arXiv:math/9806133, Bibcode:1998math......6133P, MR 1685628
- ↑ Sloane, N. J. A. (ed.). "Sequence A002559 (Markoff (or Markov) numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-13.
- ↑ Sloane, N. J. A. (ed.). "Sequence A001599 (Harmonic or Ore numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-13.
- ↑ Sloane, N. J. A. (ed.). "Sequence A038823 (Number of primes between n*1000 and (n+1)*1000)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ↑ Stein, William A. (10 February 2017). "The Riemann Hypothesis and The Birch and Swinnerton-Dyer Conjecture". wstein.org. Retrieved 6 February 2021.
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