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... Remarks: (1), (2), (3) are well-known properties of Dedekind sums. (See [1], p. 62.) (4) through (8) are well-known properties of linear recurrences of order 2. (See [2], p. 193-194). (9) can be proved via (6) or by induction on n. THE MAIN RESULTS Theorem 1: Let {un } be a linear recurrence of orde ...
... Remarks: (1), (2), (3) are well-known properties of Dedekind sums. (See [1], p. 62.) (4) through (8) are well-known properties of linear recurrences of order 2. (See [2], p. 193-194). (9) can be proved via (6) or by induction on n. THE MAIN RESULTS Theorem 1: Let {un } be a linear recurrence of orde ...
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... In 1548-1626, Cataldi [28] proved that all perfect numbers given by Euclid’s form end in 6 or 8. In 1603, Cataldi [78] found and listed all primes ≤ 750, then proved that 217 −1 = 131071 is a prime because 131071 < 562500 = 7502 , and he could check the number with his list of primes (≤ 750) to show ...
... In 1548-1626, Cataldi [28] proved that all perfect numbers given by Euclid’s form end in 6 or 8. In 1603, Cataldi [78] found and listed all primes ≤ 750, then proved that 217 −1 = 131071 is a prime because 131071 < 562500 = 7502 , and he could check the number with his list of primes (≤ 750) to show ...
