By Ben Ayed M., El Mehdi K., Grossi M.

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35 36 Burton: The History of Mathematics: An Introduction, Sixth Edition 2. Mathematics in Early Civilizations CHAPTER Text © The McGraw−Hill Companies, 2007 2 Mathematics in Early Civilizations In most sciences one generation tears down what another has built and what one has established another undoes. In Mathematics alone each generation builds a new story to an old structure. 1 With the possible exception of astronomy, mathematics is the oldest and most continuously pursued of the exact sciences.

Then take away 10 3 of this 10, giving 9. Then this was the number first thought of. Proof. If the original number was 9, then 23 is 6, which added makes 15. Then is 5, which on subtraction leaves 10. That is how you do it. 1 3 of 15 Here the scribe was really illustrating the algebraic identity n+ 2n 3 − 1 2n n+ 3 3 − 1 10 n+ 2n 3 − 1 2n n+ 3 3 =n by a simple example, in this case using the number n = 9.

That is, he multiplied the selected expression by the odd integer n to produce 2. Nowhere is there any inkling of the technique used to arrive at the decomposition. Fractions 2/n whose denominators are divisible by 3 all follow the general rule 1 1 2 = + . 3k 2k 6k Typical of these entries is 2 15 (the case k = 5), which is given as 2 15 = 1 10 + 1 . 30 If we ignore the representations for fractions of the form 2/(3k), then the remainder of the 2/n table reads as shown herewith. 2 5 2 7 2 11 2 13 2 17 2 19 2 23 2 25 2 29 2 31 2 35 2 37 2 41 2 43 2 47 2 49 2 51 = = = = = = = = = = = = = = = = = 1 1 + 15 3 1 1 + 28 4 1 1 + 66 6 1 1 1 + 52 + 104 8 1 1 1 + 51 + 68 12 1 1 1 + 76 + 114 12 1 1 + 276 12 1 1 + 75 15 1 1 1 + 58 + 174 + 24 1 1 1 + + 20 124 155 1 1 + 30 42 1 1 1 + + 296 24 111 1 1 1 + 246 + 328 24 1 1 1 + 86 + 129 + 42 1 1 1 + + 30 141 470 1 1 + 28 196 1 1 + 34 102 1 232 1 301 2 53 2 55 2 59 2 61 2 65 2 67 2 71 2 73 2 77 2 79 2 83 2 85 2 89 2 91 2 95 2 97 2 101 = = = = = = = = = = = = = = = = = 1 1 1 + 318 + 795 30 1 1 + 330 30 1 1 1 + 236 + 531 36 1 1 1 1 + 244 + 488 + 610 40 1 1 + 195 39 1 1 1 + 335 + 536 40 1 1 1 + 568 + 710 40 1 1 1 1 + 219 + 292 + 365 60 1 1 + 308 44 1 1 1 1 + 237 + 316 + 790 60 1 1 1 1 + 332 + 415 + 498 60 1 1 + 255 51 1 1 1 1 + 356 + 534 + 890 60 1 1 + 130 70 1 1 1 + 380 + 570 60 1 1 1 + 679 + 776 56 1 1 1 1 + 202 + 303 + 606 101 43 44 Burton: The History of Mathematics: An Introduction, Sixth Edition 2.

### Asymptotic behavior of least energy solutions of a biharmonic equation in dimension four by Ben Ayed M., El Mehdi K., Grossi M.

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