Unexpected biases in the distribution of consecutive primes

[PawnProver44]

91432347160754938974644339625872297 and 91432347160754938974644339625872437 are two consecutive primes both ending in 7. This only took me 3 queries to generate from the prime 91432347160754938974644339625872159. Doesn't seem like a 25% chance to me. because looking at the first prime I found starting with 7, then next prime also started with 7:

NextPrime[91432347160754938974644339625872159] = 91432347160754938974644339625872203

NextPrime[91432347160754938974644339625872203] =
91432347160754938974644339625872297

Next Prime[91432347160754938974644339625872297] =
91432347160754938974644339625872437

[science_man_88]

Originally Posted by PawnProver44

91432347160754938974644339625872297 and 91432347160754938974644339625872437 are two consecutive primes both ending in 7. This only took me 3 queries to generate from the prime 91432347160754938974644339625872159. Doesn't seem like a 25% chance to me. because looking at the first prime I found starting with 7, then next prime also started with 7:

NextPrime[91432347160754938974644339625872159] = 91432347160754938974644339625872203

NextPrime[91432347160754938974644339625872203] =
91432347160754938974644339625872297

Next Prime[91432347160754938974644339625872297] =
91432347160754938974644339625872437

but given a random number (what primes were expected to act like) ending in one of 4 endings what are the odds it ends in a specific ending 1/4 =25% edit:so in theory the next prime if acting random should have a 25% each time of having the next prime end in the same digit.

[PawnProver44]

Originally Posted by science_man_88

but given a random number (what primes were expected to act like) ending in one of 4 endings what are the odds it ends in a specific ending 1/4 =25% edit:so in theory the next prime if acting random should have a 25% each time of having the next prime end in the same digit.

Actually, Primes ending in 3 or 7 have a much higher chance (About 70-80%) than primes ending in 1 or 9 (about 20-30%).

[science_man_88]

Originally Posted by PawnProver44

Actually, Primes ending in 3 or 7 have a much higher chance (About 70-80%) than primes ending in 1 or 9 (about 20-30%).

all I was pointing out is that two primes ( even if consecutive) according to the old theory had a 25% chance of matching the last digit.

[CRGreathouse]

You're looking at 35-digit numbers with base \(q=10.\) The expected frequency of consecutive primes both ending with \(a\in\{1,3,7,9\}\) is
\[\frac{1}{4}\left(1 + c_1(10;(a,a))\frac{\log\log x}{\log x} + O\left(\frac{1}{\log x}\right)\right)\]
with
\[c_1(10; (a,a)) = \frac{4}{2}\left(\frac{1}{4} - 1\right) = -\frac{3}{2}\]
so you expect about
\[\frac14\left(1 - \frac32\frac{\log\log(10^{35})}{\log(10^{35})}\right) \approx 22.96\%\]
of the pairs to have the same last digit.

I checked the region \(10^{35}\pm10^6\) and found 23.1% which is pretty good agreement.

---

If someone works out the \(c_2\) term let me know... I haven't worked through the definitions for \(S_0^c,\ B_q,\) or the various characters \(\chi,\ \overline{\chi},\ \chi^*,\) etc.

[CRGreathouse]

Originally Posted by PawnProver44

Actually, Primes ending in 3 or 7 have a much higher chance (About 70-80%) than primes ending in 1 or 9 (about 20-30%).

No. Both are 25% asymptotically by the PNT in AP -- this has been known since the end of the 19th century.

If you're talking about the Chebyshev bias, Rubinstein & Sarnak showed (under the GRH and GSH) that it's \(25\%(1+x^{-0.5+o(1)}).\)

[only_human]

John Baez just brought up a previous paper from 1999

Andrew Odlyzko, Michael Rubinstein, and Marek Wolf, Jumping champions, Experimental Mathematics 8 (1999), 107--118, freely available at http://projecteuclid.org/euclid.em/1047477055

I mention it here because it says the most common gaps between consecutive primes are primorials and the heuristics from that paper also rely on the Hardy-Littlewood prime k-tuples conjecture:

It's not extremely easy to read, so if you want to grapple with these issues, learn about the Hardy-Littlewood prime k-tuples conjecture:

http://mathworld.wolfram.com/k-TupleConjecture.html

It gives a formula for the density of 'constellations' of primes of any shape: for example, triples of primes (n,n+2,n+40). This conjecture lies behind the surprising discovery about primes in my previous post, and also this discovery here.

So I'm not attempting to blather beyond my horizons other to wonder if in considering both papers there isn't any hint there to help tease out additional bias constraints.

[science_man_88]

Originally Posted by only_human

John Baez just brought up a previous paper from 1999

I mention it here because it says the most common gaps between consecutive primes are primorials and the heuristics from that paper also rely on the Hardy-Littlewood prime k-tuples conjecture:

So I'm not attempting to blather beyond my horizons other to wonder if in considering both papers there isn't any hint there to help tease out additional bias constraints.

I'm not well read in the theory, but that makes the bias make sense since all primorials for numbers greater than 4 ( or if primorial(x) means multiply the first x primes together, for x>2) have 2 and 5 as factors ( aka they end in 0 leading to the same last digit) .

[Kjetillll]

"If you have a prime that ends in a 7, what's the probability that the next prime ends in a 7? I would still expect the answer to be close to 25%"

Me to at first.

But if you think about it more, the gaps between two adjacent primes are not equally distributed. To have two neighbours both end in 7 requires a gap of 10, 20, 30 and so on. And as you can see below those gaps are not the most frequent.
Gap frequency:

<code>
2: 124085 ============================================
4: 124547 ============================================
6: 223695 ================================================================================
8: 100143 ===================================
10: 129451 ==============================================
12: 168665 ============================================================
14: 93866 =================================
16: 70808 =========================
18: 129276 ==============================================
20: 71818 =========================
22: 61809 ======================
24: 94213 =================================
26: 45336 ================
28: 49587 =================
30: 89678 ================================
32: 28735 ==========
34: 29823 ==========
36: 49734 =================
38: 23635 ========
40: 28118 ==========
42: 41904 ==============
44: 17104 ======
46: 14915 =====
48: 26093 =========
50: 15064 =====
52: 11501 ====
54: 19158 ======
56: 10008 ===
58: 8622 ===
60: 17669 ======
62: 5659 ==
64: 5865 ==
66: 10795 ===
68: 4374 =
70: 6232 ==
72: 6379 ==
74: 3327 =
76: 2896 =
78: 5451 =
80: 2811 =
82: 1988
84: 4229 =
86: 1503
88: 1535
90: 3151 =
92: 1067
</code>

[only_human]

via h/t David Eppstein on Google+: "Brian Hayes digs deeper into the recent discovery of correlations in the moduli of consecutive primes.
Prime After Prime

As a matter of fact, when Lemke Oliver and Soundararajan announced their findings, the response was surprise verging on incredulity. Erica Klarreich, writing in Quanta, cited the reaction of James Maynard, a number theorist at Oxford:

Evidently that was a common reaction. Evelyn Lamb, writing in Nature, quotes Soundararajan: “Every single person we’ve told this ends up writing their own computer program to check it for themselves.”

Well, me too! For the past few weeks I’ve been noodling away at lots of code to analyze primes mod m. What follows is an account of my attempts to understand where the patterns come from. My methods are computational and visual more than mathematical; I can’t prove a thing. Lemke Oliver and Soundararajan take a more rigorous and analytical approach; I’ll say a little more about their results at the end of this article.

[0PolarBearsHere]

I wonder how this would extend to looking at the last two digits of prime numbers. Both as a single 10a+b, but also as a similar investigation to the single last digit, but with two inputs.
So with the first 47 would be different to 74, but with the second they'd be the same.