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129 is the sum of the first ten prime numbers.It is the smallest number that can be expressed as a sum of three squares in four different ways: + +, + +, + +, and + +. 129 is the product of only two primes, 3 and 43, making 129 a semiprime.
We prove associativity by first fixing natural numbers a and b and applying induction on the natural number c. For the base case c = 0, (a + b) + 0 = a + b = a + (b + 0) Each equation follows by definition [A1]; the first with a + b, the second with b. Now, for the induction. We assume the induction hypothesis, namely we assume that for some ...
For example, the sum of the first n natural numbers can be denoted as ∑ i = 1 n i {\displaystyle \sum _{i=1}^{n}i} For long summations, and summations of variable length (defined with ellipses or Σ notation), it is a common problem to find closed-form expressions for the result.
Sum of Natural Numbers (second proof and extra footage) includes demonstration of Euler's method. What do we get if we sum all the natural numbers? response to comments about video by Tony Padilla; Related article from New York Times; Why –1/12 is a gold nugget follow-up Numberphile video with Edward Frenkel
A cluster prime is a prime p such that every even natural number k ≤ p − 3 is the difference of two primes not exceeding p. 3, 5, 7, 11, 13, 17, 19, 23, ... (OEIS: A038134) All odd primes between 3 and 89, inclusive, are cluster primes. The first 10 primes that are not cluster primes are: 2, 97, 127, 149, 191, 211, 223, 227, 229, 251.
In 1713, Jacob Bernoulli published under the title Summae Potestatum an expression of the sum of the p powers of the n first integers as a (p + 1)th-degree polynomial function of n, with coefficients involving numbers B j, now called Bernoulli numbers:
The first ordinal number that is not a natural number is expressed as ω; this is also the ordinal number of the set of natural numbers itself. The least ordinal of cardinality ℵ 0 (that is, the initial ordinal of ℵ 0 ) is ω but many well-ordered sets with cardinal number ℵ 0 have an ordinal number greater than ω .
A list of articles about numbers (not about numerals). Topics include powers of ten, notable integers, prime and cardinal numbers, and the myriad system.