\\ ~~~~ for oeis edits _-_

watch DYK 9,12,28&29

http://en.wikipedia.org/wiki/Portal:Mathematics/Did_you_know/9 Millenium

http://en.wikipedia.org/wiki/Portal:Mathematics/Did_you_know/12 largest prime

http://en.wikipedia.org/wiki/Portal:Mathematics/Did_you_know/28

xmas 2013 distinguishing FP from primes.

for consec integers, (a^2-b^2)=(a+b). *Andrica's conjecture* says a-b<sqrt(a)+sqrt(b) where a,b consec primes

for consec composites, either get 2 or 1 (2 iff skip an intervening odd (prime)). Eliminating all evens, result is the sequence 9,15,21,25,27,33,35,39,45,49,51,55,57,...

from 17,29,31,41,43,53,67,71,79,89,97,101,103,109,113, no hit. These are from consec composites where (a+b) is prime.

However, primes MISSED by the procedure are A079149, 3,5,7,11,13,23,37,47,59,61,73,83,107,...

**Proizvolov's identity ref http://en.wikipedia.org/wiki/Proizvolov%27s_identity**

** **

*x*^{a} − *y*^{b} = 1 only soln in natl numbers is 3^2-2^3 (Catalan)

**Zeckendorf's theorem ref http://en.wikipedia.org/wiki/Zeckendorf%27s_theorem **

**A conjecture by Sun:**

Every odd integer n ≥ 5 can be written in the form **p + x(x + 1)** where p is a prime and x a positive integer.

Ref http://Z. W. Sun, On sums of primes and triangular numbers, Journal of Combinatorics and Number Theory 1(2009), no. 1

in other words, every odd n>5 is a prime plus product of 2 consecutive integers. Thus, every prime n>5 = a prime plus prod of 2 consecutive integer.

Related, via Lemoine every odd n>5 = p+2q where p,q are primes (see A046927)

With primes, (2) useful properties (does incl False hits)

Mod(P,30) must reside in (1,7,11,13.17,19,23,29)

Fermat's Little Theorem Mod( (2^(p-1), P) =1

I have implemented this in Pari/GP, though size limitation seems ~ 525000000 (5E8). %correct >99.9

Misc approx

Pi/Euler~ 2*e ie 5.443 vs 5.437

Euler*(Phi+e) ~ Pi/e + e/sqrt(2) ie 3.0802 vs 3.0778

Erdos-Borwein ~ (sqrt(2)/Euler+Pi/e )-2 ie 1.6067 vs 1.6058

Mill's constant*2 -1 ~ Phi ie 1.613 vs 1.618

I am producing prime sequences from concatenated sequences like Mill's prime. Submissions due shortly, 2+

**** versions of 1 (9/2/2016) ***

It is well known that 0.999....equals 1. It is interesting to think further.

In the same vein, 1.000...1 should also=1. Let a=0.999.... and b=1/a=1.000....1

We know a *1/a =1. Of course a-a=0. This implies 1.000...1 - 0.999... should =0

Generally alternate forms of the same number in the same form is troublesome.

However, the basic laws of our math systems doesn't require or presume such forms can't exist.

1/0.999 =1.001001001...

1/0.9999= 1.000100010001...

etc

Similarly, (1+a)-(1-a)=2a eg (1+0.20)-(1-0.20) = 1.2-0.80=0.40=2*0.20. Note 1/0.8 != 1.20

So, with a=0.000...1, 2a=0.000...2

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