proof that exceeds the product of the primes up to
Statement. (Erdős & Surányi) Given the prime counting function and notating the th prime as , the inequality
is always true for any nonnegative . In other words, the th power of 4 is greater than the primorial .
Note: is the integer nearest to the real number that’s not greater than , while is the nearest integer to that’s not smaller than .
Proof. It takes very little computational effort to verify this is true for set to 1 or 2. For larger we can high-ball the primorial as
(that is, twice the product of the odd numbers from 1 to ) and rephrase as
The first factor is obviously the same for both expressions, both iterators have the same start value, but cleary the iterator end values satisfy the inequality as long as . The first expression being iteratively multiplied clearly increases, but the second expression being iteratively multiplied is static (being a 2). This line of thought has failed to yield the desired result.
What if instead we divide both expressions by 2 and take their base 2 logarithms? Obviously, for half ,
giving us the sequence of odd numbers 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, etc. Not quite so obviously, for half our primorial high-ball, the base 2 logarithm is
(where is the natural logarithm and is the double factorial), giving us the sequence (to 5 decimal places) 0, 1.58496, 1.58496, 3.90689, 3.90689, 6.71425, 6.71425, 9.88417, 9.88417, 13.3436, 13.3436, 17.044, 17.044, 20.9509, 20.9509, 25.0384, 25.0384, 29.2863, 29.2863, etc.
Unfortunately, both of these strategies are doomed to failure because eventually our high-ball for the primorial overtakes . What Erdős and Surányi do instead is rephrase as and then construct the binomial coefficient and showing the relationship of this to .
- 1 Paul Erdős & János Surányi Topics in the theory of numbers New York: Springer (2003): 5.11
|Title||proof that exceeds the product of the primes up to|
|Date of creation||2013-03-22 17:04:48|
|Last modified on||2013-03-22 17:04:48|
|Last modified by||PrimeFan (13766)|