Search results
Results from the WOW.Com Content Network
Proof without words of the arithmetic progression formulas using a rotated copy of the blocks. An arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference from any succeeding term to its preceding term remains constant throughout the sequence. The constant difference is called common difference of that ...
The case = coincides with that of the calculation of the arithmetic series, the sum of the first values of an arithmetic progression. This problem is quite simple but the case already known by the Pythagorean school for its connection with triangular numbers is historically interesting:
For example, (), is a well-formed formula of second-order arithmetic that is arithmetical, has one free set variable X and one bound individual variable n (but no bound set variables, as is required of an arithmetical formula)—whereas (<) is a well-formed formula that is not arithmetical, having one bound set variable X and one bound ...
For example, 6n + 1 produces the same primes as 3n + 1, while 6n + 5 produces the same as 3n + 2 except for the only even prime 2. The following table lists several arithmetic progressions with infinitely many primes and the first few ones in each of them.
For example, the sequence,,,,, … is not an arithmetic progression, but is instead generated by starting with 17 and adding either 3 or 5, thus allowing multiple common differences to generate it. A semilinear set generalizes this idea to multiple dimensions – it is a set of vectors of integers, rather than a set of integers.
Roth's theorem on arithmetic progressions (infinite version): A subset of the natural numbers with positive upper density contains a 3-term arithmetic progression. An alternate, more qualitative, formulation of the theorem is concerned with the maximum size of a Salem–Spencer set which is a subset of [ N ] = { 1 , … , N } {\displaystyle [N ...
In number theory, primes in arithmetic progression are any sequence of at least three prime numbers that are consecutive terms in an arithmetic progression. An example is the sequence of primes (3, 7, 11), which is given by a n = 3 + 4 n {\displaystyle a_{n}=3+4n} for 0 ≤ n ≤ 2 {\displaystyle 0\leq n\leq 2} .
When this equation is satisfied, both sides of the equation equal the congruum. Fibonacci solved the congruum problem by finding a parameterized formula for generating all congrua, together with their associated arithmetic progressions. According to this formula, each congruum is four times the area of a Pythagorean triangle.