Mills' Constant?

Question

it is known that there exist real positive numbers θ such that the integer part of θ^3^n is a prime number, for all positive integers n. The least such number is called Mills' constant and is approximately

1.30637788386308069046...

http://en.wikipedia.org/wiki/Mills%27_constant

My Question:
Is there any reason why Mills has chosen the number 3 in his formula.
It's obvious to me that one can chose any real number k > 1 so that Int(θ^k^n) gives also prime numbers for each n.
Of course we will get a different θ if k is not 3.

Answer 1

If Θ is raised to an even power, then it will always (for some power) generate a perfect square (or product of perfect squares) and hence will not always be prime.

Actually, any odd value would work, but 3 is the smallest odd prime.


Doug

Answer 2

Here is the answer:

he least possible value for Mills constant (usually called "the Mills Constant") begins as follows:
1.3063778838 6308069046 8614492602 6057129167 8458515671 3644368053 7599664340 5376682659 8821501403 7011973957 0729696093 8103086882 2388614478 1635348688 7133922146 1943534578 7110033188 1405093575 3558319326 4801721383 2361522359 0622186016 1085667905 7215197976 0951619929 5279707992 5631721527 8412371307 6584911245 6317518426 3310565215 3513186684 1550790793 7238592335 2208421842 0405320517 6890260257 9344300869 5290636205 6989687262 1227499787 6664385157 6619143877 2844982077 5905648255 6091500412 3788524793 6260880466 8815406437 4425340131 0736114409 4137650364 3793012676 7211713103 0265228386 6154666880 4874760951 4410790754 0698417260 3473107746 7757406400 7810935083 4214374426 5420408531

Source: The PRIME GLOSSARY

Answer 3

Because 3 is the smallest odd prime.

I agree with a previous post.

Answer 4

3 was chosen because it is occipedal.

Answer 5

this is false how can n be irrational