Legendre speculated based on a meticulous examination of prime tables available to him in 1808 that: Define the prime counting function \( \pi(x) \) as the number of primes less than or equal to the positive real number \( x, \) so the remarks above can be reframed as: There are 25 primes between 1 and 100, and the primes do seem to thin out on average as one proceeds to greater values. There are fourteen primes between 500 and 600, for example, with 521 being prime, 522 not prime (composite), 523 prime, followed by 17 composites in a row before the next prime at 541. The density of the primes within the stream of integers oscillates wildly and seemingly without pattern. The first five values of \( m \) here actually are prime, but the last one is not, its factorization producing two new primes beyond \( 2, 3, 5, 7, 11, 13 \) - namely, \( 59 \) and \( 509. Note that (despite a promising start), \( m = p_1 \cdot p_2 \cdot p_3 \cdots p_n + 1 \) is not necessarily prime itself, it just has a prime factor other than \( p_1, p_2, \ldots p_n: \)Ģ \cdot 3 \cdot 5 \cdot 7 \cdot 11 + 1 &= 2311\\Ģ \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13 + 1 &= 30031 = 59 \cdot 509.\\ This proof assumes that every natural number has at least one prime divisor, but this can be filled in. It follows that \( p_1 \) cannot divide \( m \) and similarly none of the \( p_j \) can divide \( m, \) all of whose prime divisors must therefore be other than \( p_1, p_2, \cdots p_n. \) That is, \( p_1 | 1 \) and therefore \( p_1 = 1, \) which is not a prime number. Then every prime divisor of \( m = p_1 \cdot p_2 \cdot p_3 \cdots p_n + 1 \) is different from \( p_1, p_2, \ldots p_n, \) so there is at least one more prime. Suppose \( p_1, p_2, \ldots p_n \) represent the first \( n \) prime numbers: \( p_1 = 2, p_2 = 3, p_3 = 5, \) and so on. The modern proof goes like this: Theorem. Like all of Euclid, the proof is geometrical, with line segments representing numbers, but it's valid and recognizable.
Euclid proved that there are infinitely many prime numbers in 300 BC in Book IX, Proposition 20 of the Elements. The key fact about the primes is that every natural number can be written as a product of primes, and the product is unique up to the order of the factors. These are the counting numbers having no divisors other than one and themselves: Since zero is false, I suppose the "!= 0" parts could be omitted from the conditionals, but I think it's easier to read with them in.The prime numbers have been an object of fascination for a long time.
The integer division used to calculate m will have to be changed for Python 3. m is a little tricky it's the number of multiples of i that are greater than i**2 and less than or equal to n.
#List of prime numbers under 10000000 code
The three terms that define the slice are the same as the three terms that define the range for j in the "Alex" code, which is why I think the code is still readable. The slice assignment seems to be the big time saver. The code below runs about 50% faster than the "Alex" code and isn't too obscure, I think: nroot = int(sqrt(n))
Twice as fast? In my benchmarks (on an Intel iMac with n=1,000,000, n=10,000,000, and n=20,000,000), the recipe code is only about 10% faster than the "Alex" code, although its memory use is probably much smaller. Def primes ( n ): if n = 2 : return elif n < 2 : return s = range ( 3, n + 1, 2 ) mroot = n ** 0.5 half = ( n + 1 ) / 2 - 1 i = 0 m = 3 while m <= mroot : if s : j = ( m * m - 3 ) / 2 s = 0 while j < half : s = 0 j += m i = i + 1 m = 2 * i + 3 return + print primes ( 13 ) print primes ( 3000 ) - we get.
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