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33 is the 21st composite number, and 8th distinct semiprime (third of the form where is a higher prime). [1] It is one of two numbers to have an aliquot sum of 15 = 3 × 5 — the other being the square of 4 — and part of the aliquot sequence of 9 = 3 2 in the aliquot tree (33, 15, 9, 4, 3, 2, 1).
7 is the only number D for which the equation 2 n − D = x 2 has more than two solutions for n and x natural. In particular, the equation 2 n − 7 = x 2 is known as the Ramanujan–Nagell equation. 7 is one of seven numbers in the positive definite quadratic integer matrix representative of all odd numbers: {1, 3, 5, 7, 11, 15, 33}. [19] [20]
[33] [b] There are problems with Tacitus's work, however, as it is unclear how much he really knew about the Germanic peoples he described and because he employed numerous topoi dating back to Herodotus that were used when describing a barbarian people. [35]
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