By Jeffrey Stopple
This undergraduate-level advent describes these mathematical houses of best numbers that may be deduced with the instruments of calculus. Jeffrey Stopple can pay specified realization to the wealthy background of the topic and historic questions about polygonal numbers, ideal numbers and amicable pairs, in addition to to the real open difficulties. The end result of the publication is a quick presentation of the Riemann zeta functionality, which determines the distribution of best numbers, and of the importance of the Riemann speculation.
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Extra resources for A Primer of Analytic Number Theory: From Pythagoras to Riemann
8. You now know enough to compute sums of any mth power in closed form. Show that 15 + 25 + · · · + n 5 = (2n 2 + 2n − 1)(n + 1)2 n 2 /12. 16) You can ﬁnd out more about Stirling numbers in Graham, Knuth, and Patashnik, 1994. As with the polygonal numbers, once we have a closed-form expression, there seems to be nothing left to say. 13) misses one case. There is no factorial power whose difference is n −1 . In other words, n −1 is not a factorial power. ) So we make a deﬁnition instead, deﬁning the nth harmonic number to be Hn = 1≤k≤n 1 1 1 = 1 + + ··· + .
Ptkt are all relatively prime, (n) = p1k1 p2k2 . . ptkt = (k1 + 1)(k2 + 1) . . (kt + 1), and similarly for (n). Theorem (Euclid). If p is a prime number such that Mp = 2 p − 1 is also a prime, then n = 2 p−1· Mp is a perfect number. Proof. To show s(n) = n, we show that (n) = s(n) + n is just 2n. By the previous theorem, (n) = (2 p−1 M p ) = (2 p−1 )(M p ) = 2 p − 1 M 2p − 1 = (2 p − 1)(M p + 1), 2 − 1 Mp − 1 38 2. Products and Divisibility but 2 p − 1 is just M p and therefore M p + 1 is just 2 p .
C. Problem 79 says, “There are seven houses. Each house has seven cats. Each cat catches seven mice. Each mouse eats seven ears of spelt [a grain related to wheat]. Each ear of spelt produces seven hekats [a bulk measure]. ” Use the Geometric series to answer this, the oldest known mathematical puzzle. Archimedes, too, knew of the Geometric series. ). Archimedes is better known for his beautiful theorems on area and volume in geometry than for his work in number theory. However, the Geometric series and other series, as we will see, are vital in number theory.
A Primer of Analytic Number Theory: From Pythagoras to Riemann by Jeffrey Stopple