# The Book of Squares: An Annotated Translation into Modern by Leonardo Pisano Fibonacci

By Leonardo Pisano Fibonacci

The booklet of Squares by means of Fibonacci is a gem within the mathematical literature and probably the most vital mathematical treatises written within the center a long time. it's a choice of theorems on indeterminate research and equations of moment measure which yield, between different effects, an answer to an issue proposed via grasp John of Palermo to Leonardo on the court docket of Frederick II. The ebook was once committed and offered to the Emperor at Pisa in 1225. relationship again to the thirteenth century the ebook shows the early and persisted fascination of fellows with our quantity process and the connection between numbers with distinctive houses comparable to leading numbers, squares, and bizarre numbers. The devoted translation into glossy English and the statement through the translator make this booklet available to expert mathematicians and amateurs who've consistently been intrigued via the entice of our quantity procedure.

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Fermat, in a letter to Carcavi [79, Vol. II, p. 433], indicated that his proof (like so many others of his) was based on the method of descent, whose application in this case, he added, required another new idea. Euler, in a letter of 1730 to Goldbach [54], mentioned that he could not prove Theorem 2, the real difficulty residing in showing that numbers of the form n 2 + 7 are sums offour squares. Indeed, 7 cannot be written as a sum of three squares, while if a = If=l then obviously n 2 + a is a sum of four squares.

Sketch of Jacobi's Proof of Theorem 4 0 = 1 1 00 -6cotz- + -2 L nUn 1 2 n=1 = 1 0 -6 cot 2 1 2 L2 + 00 L uk(l k=1 as claimed. 12), and T2 = = + k) L Uk 1 + Uk - -2 coskO k=1 00 ( 1 00 + Uk) COS kO + -2 L k=1 ku k(l - cos kO) n12, we obtain ! 13), we obtain which proves Lemma 3 and completes the proof of Theorem 4. §8. 14) where (see Chapter 1) only the last equality still needs justification. Assuming it, we observe that for m odd, mxm L m odd m 1 + (- 1) 00 m x = L mxm(1 m odd 00 + xm + x2m + ...

We shall not need the complete theory of genera-not even their precise definition. , [105, §48, Satz 145, §53]. , [105, §48, Satz 145, §53]), that each of the 9 genera of a given discriminant contains the same number k = h/g of equivalence classes. If the discriminant D = b2 - 4ae of the form ax2 + bxy + ey2 is divisible by t distinct primes, then 9 = 2t - 1 , except for -D = 4n, with n == 3 (mod 4), when 9 = 2t - 2 , and for -D = 4n, n == (mod 8), when 9 = 2t-cases that will not concern us. This is essentially equivalent to Gauss's rule of counting only the odd prime divisors of n.