the space stations of the cross

Isaiah 53:2 — “He had no beauty or majesty to attract us to him, nothing in his appearance that we should desire him.”

only the dead have seen the end of war

 

Bertrand's Postulate and a conjecture for any three consecutive primes - key to formula for the next largest prime?

The postulate states that for any integer n >1, there is at least one prime p such that n < p < 2n. 

In a similar way, I conjecture that for any three consecutive primes p, q, and r with p < q < r:

(p + q)/r = 2 (approximately) 

For large values of p, q, and r, the expression approaches 2. 

Checking this conjecture for some sample values:

(991 + 997)/1009 = 1.97

(9967 + 9973)/10007 = 1.992

(99971 + 99989)/99991 = 1.999 

A crude approximation for the next largest prime r would then be that r is slightly greater than (p + q)/2. 

(99971 + 99989)/2 = 99980

(99991 - 99980)/99980 = 0.00011 or about 0.011%

Because of the Goldbach Conjecture, p + q will always be even and thus (p + q)/2 will always be an integer. 

If we wanted to find the next prime larger than 99991, a good initial guess would be 1.00011*99991 = 100002. That isn't prime but 100003 is. 

So if you have three consecutive primes, you can get a very good estimate of the next largest prime

by calculating the percent difference as shown above. 

Euler's prime formula, the Bunyakovsky Conjecture, and n^2 + n + 1

Euler's prime formula of n^2 + n + 41 produces prime numbers for all integers n from 1 to 40. 

If 41 is substituted for another larger prime, it produces a larger proportion of the first 1,000 primes.

Derpetology: The Archimedes, Ulam, and Sacks Spirals, Euler's Prime Formula, and the Non-random Distribution of Prime Numbers

n^2 + n + 1 produces prime numbers for all n that are congruent to n = 0 or 2 in modulo 6. 

Bunyakovsky conjecture - Wikipedia

Derpetology: spirals, primes, conjectures, and theorems

If we understood primes perfectly, we'd understand pi perfectly. Every prime is a little slice of pi. 

It all fits together somehow. 

he came in peace

 

John 8:23 NIV - But he continued, “You are from - Bible Gateway

the tree of silence bears the fruit of peace

 

-Counsels and Maxims of Schopenhauer

diablo