**INTRODUCTION**

The prime numbers are numbers which are divisible only by itself and 1. This means that all primes have two factors, so for that reason, “1” is not prime.

**OBSERVATIONS**

The first 100000 integers seems to have the greatest density of prime numbers. 9592 primes were found there, meaning on average, nearly one in 10 of the first 100000 integers were prime.

Dissecting this interval further, for the first 1000 contiguous integers there are 168 primes:

INTERVAL # OF PRIMES FRACTION 1-100 25 0.25 101-200 21 0.21 201-300 16 0.16 301-400 16 0.16 401-500 17 0.17 501-600 14 0.14 601-700 16 0.16 701-800 14 0.14 801-900 15 0.15 901-1000 14 0.14 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ TOTAL 168 0.168

we see we’re already in trouble. While the first 200 integers clearly show dominance in having the most primes, with the first 100 integers having the most of all, the remainder fall into some kind of pseudo-random torpor, with some intervals higher, and some lower. 16 primes occur in three of the intervals, and 14 occur in another three.

In this interval, it is difficult to figure out if the numbers are meandering up or down. Let’s look at the second thousand:

1001-1100 16 0.16 1101-1200 12 0.12 1201-1300 15 0.15 1301-1400 11 0.11 1401-1500 17 0.17 1501-1600 12 0.12 1601-1700 15 0.15 1701-1800 12 0.12 1801-1900 12 0.12 1901-2000 13 0.13 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ TOTAL: 135 0.135

Is this a downward trend? We observe the first 10 thoudsands:

1-1000 168 0.168 1001-2000 135 0.135 2001-3000 127 0.127 3001-4000 120 0.120 4001-5000 119 0.119 5001-6000 114 0.114 6001-7000 117 0.117 7001-8000 107 0.107 8001-9000 100 0.100 9001-10000 112 0.112 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ TOTAL: 1219 0.1219

A major jump upward is observed in the last interval, but it does not get to the level of the first 4 intervals. This is noticed nearly universally, and is reproduced here. If you look at the number of primes in regularly-spaced intervals of contiguous sequences of integers with each interval the same size, you tend to see a general declining trend, followed by a seemingly random meanindering of numbers of primes going up and down in number. This observation appears to be independent of

interval size.

In the interval 99001 to 100000, the last 1000 numbers in the interval, we observe that there are 86 primes. Clearly, this is below what we have just observed in the first 10 intervals of 1000, but in the interval 97001-98000, we observe only 82 primes, where we had expected a slightly greater number than 86. The number of primes once again are returning to a similar pseudo-random torpor observed earlier.

I define “pseudo” randomness as frequencies which go on a downward trend in an unpredictably meandering way. True randomness would have frequencies going all over the place.

So, if intervals of size “p” were used, the first p integers would contain the most primes; while there would be a downward trend for integers from p+1 to 2p; 2p+1 to 3p, and so on. By some middle interval kp+1 to (k+1)p, there would occur a relatively low number of primes in some pseudo random torpor. By the time we arrived at interval np+1 to (n+1)p, the frequency of primes become noticeably fewer. I am afraid I am not yet in a position to define the word “noticeably”. You just have to notice it.

But the word “noticeably” might imply that there is an upper limit of frequencies which it does not attain. That is, if “e” was the upper limit of a predicted frequency by interval np+1 to (n+1)p, we need to set the number so as to guarantee to ourselves that it does not rise above the limit. As a first approximation, I am willing to set e, if 2 divides n, as being the same for the interval (n/2)p+1 to ((n/2)+1)p, or the interval half-way to the last one. You obviously can be more restrictive than that, but at least you can see from my numbers that this clearly works out.

**Contiguous Intervals of 100000**

Taking the whole interval of the first 100000 positive integers contiguously, we find there are 9592 primes, far and away more than any other such interval observed for 100000 integers. The second interval from 100001 to 200000, has 8392 primes. Going from 1 billion, the interval has 4832 primes. While it is not clear if we are returning to torpor, we can see at least that the number of primes are decreasing.

The strongest proof I have observed of a decline returning to a chaos of up-and-down numbers of primes, was to see that these observations were even consistent when I counted 100,000 contiguous integers, and jumping 10^50 integers and counting another interval of 100,000 from there. I kept this up under Maple 9.5 up to 10^2000. The last intervals from 10^1700 took the longest to count (well, _you_ try checking 100000 integers which are each 1700 digits long, and see how long it takes you!). It is still cranking away, and there are 4 intervals left. In all, it will probably take 9 hours on a dual-core processor, so on the optimistic side, I have about 3 hours remaining.

The results so far have been enough to more than indicating a trend:

100000 FROM # OF PRIMES EQUAL OR GREATER(*) 10^0 9592 10^50 895 10^100 407 10^150 274 10^200 216 10^250 165 10^300 143 10^350 129 10^400 115 10^450 96 10^500 81 10^550 71 10^600 78 * 10^650 65 10^700 68 * 10^750 59 10^800 52 10^850 57 * 10^900 57 * 10^950 45 10^1000 39 10^1050 38 10^1100 43 * 10^1150 33 10^1200 38 * 10^1250 43 * 10^1300 21 10^1350 32 * 10^1400 28 10^1450 28 * 10^1500 26 10^1550 35 * 10^1600 32 10^1650 28 10^1700 28 * 10^1750 29 * 10^1800 27 10^1850 23

As has been stated by other writers, prime numbers appear to have a pattern, but the pattern has eluded us. It seems the pattern of decreasing numbers of primes as integers increase by many orders of magnitude seems to be elusive indeed, appearing pseudo-random, but with a general downward trend. One might desire to regress the numbers of primes per 100,000 integers to a curve, but I am not sure that would tell us anything meaningful by way of a pattern.