Squared Triangular Number
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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In number theory, the sum of the first n cubes is the square of the nth triangular number. That is, sum_{i=1}^{n} i^3 = Bigl(sum_{i=1}^{n} iBigr)^2. This identity is sometimes called Nicomachus''s theorem. Stroeker (1995), writing about Nicomachus''s theorem, claims that "every student of number theory surely must have marveled at this miraculous fact". While Stroeker''s statement may perhaps be a poetic exaggeration, it is true that many mathematicians have studied…mehr

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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In number theory, the sum of the first n cubes is the square of the nth triangular number. That is, sum_{i=1}^{n} i^3 = Bigl(sum_{i=1}^{n} iBigr)^2. This identity is sometimes called Nicomachus''s theorem. Stroeker (1995), writing about Nicomachus''s theorem, claims that "every student of number theory surely must have marveled at this miraculous fact". While Stroeker''s statement may perhaps be a poetic exaggeration, it is true that many mathematicians have studied this equality and have proven it in many different ways. Pengelley (2002) finds references to the identity in several ancient mathematical texts: the works of Nicomachus in what is now Jordan in the first century CE, Aryabhata in India in the fifth century, and Al-Karaji circa 1000 in Persia. Bressoud (2004) mentions several additional early mathematical works on this formula, by Alchabitius (tenth century Arabia), Gersonides (circa 1300 France), and Nilakantha Somayaji (circa 1500 India); he reproduces Nilakantha''s visual proof.