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For instance, the sequence 5, 7, 9, 11, 13, 15, . . . is an arithmetic progression with a common difference of 2. If the initial term of an arithmetic progression is and the common difference of successive members is , then the -th term of the sequence is given by
For example, the sequences (2, 3, 4) and (2, −3, 4) are trivial Büchi sequences. A nontrivial Büchi sequence of length three is given for example by the sequence (0, 7, 10), as it satisfies 10 2 − 2·7 2 + 0 2 = 2, while 0 2, 7 2 and 10 2 are not consecutive squares.
In mathematics, a harmonic progression (or harmonic sequence) is a progression formed by taking the reciprocals of an arithmetic progression, which is also known as an arithmetic sequence. Equivalently, a sequence is a harmonic progression when each term is the harmonic mean of the neighboring terms.
For example, the AP-3 {3, 7, 11} does not qualify, because 5 is also a prime. For an integer k ≥ 3, a CPAP-k is k consecutive primes in arithmetic progression. It is conjectured there are arbitrarily long CPAP's. This would imply infinitely many CPAP-k for all k. The middle prime in a CPAP-3 is called a balanced prime.
The history of the problem dates back at least to Gersonides, who proved a special case of the conjecture in 1343 where (x, y) was restricted to be (2, 3) or (3, 2).The first significant progress after Catalan made his conjecture came in 1850 when Victor-Amédée Lebesgue dealt with the case b = 2.
The sector has lagged the S&P 500's rally this year, up just 1%, while the index has surged 26%. After a tough year marked by policy uncertainty and lagging fundamentals, healthcare stocks are set ...
After three years, you’d have earned $900 in interest — $300 each year — for a total of $10,900 in your account. Now let's say you invest $10,000 in an account that pays 3% compounded annually.
Taniyama's problems [3] 36-Yutaka Taniyama: 1955 Thurston's 24 questions [4] [5] 24-William Thurston: 1982 Smale's problems: 18: 14: Stephen Smale: 1998 Millennium Prize Problems: 7: 6 [6] Clay Mathematics Institute: 2000 Simon problems: 15 <12 [7] [8] Barry Simon: 2000 Unsolved Problems on Mathematics for the 21st Century [9] 22-Jair Minoro ...