7. arithmetic & number theoretic recreations a. Fibonacci numbers

121. n  7 (8), 3 (13).

122. n  1 (2, 3, 4, 5).

123. n   1 (2, 3, 4, 5).

124. n  1 (2), 0 (3), 3 (4), 4 (5).

127. 2n  3 (9), 3n  5 (11), 5n  2 (8).

Ibn al Haitam. c1000. ??NYS. Problem A 7 (= Bhaskara I). (English in Libbrecht, p. 234.) (See also: E. Wiedemann; Notiz über ein vom Ibn al Haitam gelöstes arithmetisches Problem; Sitzungsber. der phys. Soz. in Erlangen 24 (1892) 83. = Aufsätze zur Arabischen Wissenschaftsgeschichte; Olms, Hildesheim, 1970, vol. 2, p. 756.) Answer:  721.

Bhaskara II. Bijaganita. 1150. Chap. VI, v. 160 & 162. In Colebrooke, pp. 235 237 & 238 239.

V. 160. Problem C (= Brahmagupta).

V. 162. n  1 (2), 2 (3), 3 (5).

Fibonacci. 1202.

Pp. 281 282 (S: 402). Problem A 7 (= Bhaskara I). He says the answer is 301 but that one can add 420. He doesn't mention this point in later problems.

Pp. 282 283 (S: 402-403). Problems A 11, A 23, D-7, D-10, D-23. These are all first appearances of these forms, and A-23, D-10, D-23 never occur again, while A 11 only reappears in Tartaglia.

P. 304 (S: 428-429). n  2 (3), 3 (5), 4 (7); problems E & F. (See Libbrecht, pp. 236 238 for Latin and English.)

Chhin Chiu shao (= Ch'in Chiu Shao = Qin Jiushao). Shu Shu Chiu Chang (Mathematical Treatise in Nine Sections). 1247. Complete analysis. (See Libbrecht, passim. See also Mikami 65 69 and Li & Yuan (op. cit under Sun Zhi). (n    32 (83),  70 (110),  30 (135) is given by Mikami 69.)

Yang Hui. Hsü Ku Chai Ch'i Suan Fa. 1275. Loc. cit. in 7.N, pp. 151 153, problems 1 5.

1. n  2 (3), 3 (5), 2 (7) & problem E (= Sun Zi).

2. n  2 (3), 3 (5), 0 (7).

3. n  1 (7), 2 (8), 3 (9).

4. n  3 (11), 2 (12), 1 (13).

5. n  1 (2), 2 (5), 3 (7), 4 (9).

Giovanni Marliani. Arte giamata arismeticha. In codex A. II. 39, Biblioteca Universitaria de Genova. Van Egmond's Catalogue 139 dates it c1417. Described and partly transcribed by Gino Arrighi; Giuochi aritmetici in un "Abaco" del Quattrocento Il matematico milanese Giovanni Marliani. Rendiconti dell'Istituto Lombardo. Classe di Scienze (A) 99 (1965) 252 258. Prob. IV: D-7.

Pseudo-dell'Abbaco. c1440.

Prob. 114, p. 95. n   1 (2, 3, ..., 10) (C-10). Takes 10!   1.

Prob. 115, p. 96. n  1 (2, 3, ..., 10). Takes 7560 + 1 and says 75601 also works.

AR. c1450. Prob. 268, 311, 349. Pp. 120 121, 138 139, 153, 181, 228 229.

268. Divinare. Three moduli used for divination. Gives the multipliers for triples; 3, 5, 7; 2, 3, 5; 3, 4, 5; 3, 4, 7; 2, 3, 7; 2, 7, 9; 5, 6, 7; 5, 8, 9; 9, 11, 13.

311. Case 3, 5, 7 of prob. 268 = Problem E.

349. Problem A 7 (= Bhaskara I). Answer: 721.

Correspondence of Johannes Regiomontanus, 1463?-1465. Op. cit. in 7.P.1.

P. 219, letter to Bianchini, late 1463 or early 1464, question 8: n    15 (17),  11 (13),  3 (10).

P. 237, letter from Bianchini, 5 Feb 1464. Bianchini answers the above problem with 1103 and 3313 and says there are many more solutions, but he doesn't wish to spend the labour required to find more. Curtze notes that Bianchini must not have understood the general solution.

P. 254, letter to Bianchini, nd [presumably 1464]. Notes that 1103 is the smallest solution and that the other solutions are obtained by adding the product of 17, 13 and 10, namely 2210. Curtze notes that Regiomontanus clearly understood the general solution.

P. 295, letter to von Speier, nd [apparently early 1465]. Prob. 6: n    12 (23),  7 (17),  3 (10).

Benedetto da Firenze. c1465. Pp. 68 69. Problems A 7 (= Bhaskara I), D-7 (= Fibonacci). He indicates the general answers.

Muscarello. 1478. Ff. 69r-69v, p. 180. n  -1 (2, 3, 4, ..., 10) (= Pseudo-dell'Abbaco).

della Francesca. Trattato. c1480. F. 122v (261). A-7. Answer: 721. English in Jayawardene.

Chuquet. 1484.

Prob. 143. English in FHM 227, with reproduction of original on p. 226. Prob. A 7 (= Bhaskara I). Gets 301 by trial and error and says there are other solutions, e.g. 721, 519841, 90601. "Thus it appears that such questions may have several and divers responses."

Prob. 144. n  2 (3, 4, 5, 6), 0 (7). Mentioned on FHM 227, which erroneously implies n  1 (3, 4, 5, 6).

HB.XI.22. 1488. Pp. 52 53 (Rath 247). Prob. A 7 (= Bhaskara I). Answer: 721. Editor notes that 301 is the smallest solution.

Pacioli. De Viribus. c1500. Problems XXII - XXV.

Ff. 34v - 36v. XXII effecto atrovare un numero pensato non piu de 105 (XXII effect: to find a number thought of not larger than 105) = Peirani 62-64. Problem E.

Ff. 36v - 39r. XXIII. effecto atrovare un Numero pensato non piu de 315 (XXIII effect: to find a number thought of not larger than 315) = Peirani 64-67. Problem F.

Ff. 39r - 42r. XXIIII. effecto 1 n ch' partito per 2.3.4.5.6. sempre avanzi 1^o. et partito per .7. avanzi nulla (XXIIII effect: a number which divided by 2, 3, 4, 5, 6, always leaves remainder 1 and divided by seven leaves nothing) = Peirani 67-71. Problem A 7 (= Bhaskara I). Discusses general solution. Then does A-10.

Ff. 42r - 44r. XXV. effecto atrovare un Nů ch' partito in 2. avanza 1^o. in .3 . 2. in .4 . 3. in 5 . 4. in 6. 5. in 7. nulla etc (XXV effect: to find a number which divided by 2 leaves remainder 1; by 3, 2; by 4, 3; by 5, 4; by 6, 5; by 7, nothing, etc.) = Peirani 71-73. Problem D-7 (= Fibonacci). Gives general solution. Then tries to solve D-9 as 2+1*3+2*4+3*5+4*6+5*7+6*8+7*9 = 725751, but this is    9 (2,3,...,8), 0 (9). A marginal note, omitted by Peirani, is basically illegible in Uri's photo and the microfilm, but seems to be noting that the answer is divisible by 3, hence does not have remainder 2 when divided by 3. In fact, D-9 is inconsistent and unsolvable. He then considers D-11 and gets the answer 2519 and says one can determine a value whose multiples can be added to 2519 to get more solutions, but he doesn't compute this. He then examines D-23 and gets the minimal solution 4,655,851,199, which is 20 * LCM (2, 3, ..., 22) - 1. He then says that if one wants the remainder on division by 23 be other than 0, and seems to say that if one takes the same condition, then one gets 698,377,681. This is 3 * LCM (2, 3, ..., 22) + 1, and is the solution of A-23.

Tagliente. Libro de Abaco. (1515). 1541. Prob. 116, f. 57v. Woman with basket of eggs -- problem A 7 (= Bhaskara I).

Ghaligai. Practica D'Arithmetica. 1521. Prob. 26, f. 66r. Basket of eggs -- problem D 7 (= Fibonacci).

Cardan. Practica Arithmetice. 1539. Chap. 66, sections 63 & 64, ff. FF.i.v - FF.ii.r (pp. 154 155). Problems A-7 (= Bhaskara I) & D-7 (= Fibonacci).

Tartaglia. General Trattato, 1556, art. 146 150, pp. 257v 258v; art. 199, p. 264r.