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First published Feb 19.
Revised Feb. 22, 2008.

*If you think I'm out on a limb here, read this Merriam-Webster usage note.

QPP-2 is now in Sloane's:
OEIS Sequence A134969

Comprehensive Data Available

The complete data set of paired primes under 100,000 for the 500 even quadratic intervals up to 1,000, are available for download here.

There are 500 files (one for each interval and identified by the interval number). Each file comprises two parts:

First, a listing of paired primes:
-1st Prime
-2nd Prime
-1st Perfect Square
-2nd Perfect Square
-Offset from the 1st perfect square
-Q-Pair Product*
-Q-Pair Product's preceding perfect square*
-Q-Pair Product Offset from preceding perfect square*

Second, a listing of the number of prime pairs occurring for each perfect square offset.

*For a clue as to why one might care about the last 3 items, see Factoring a Paired-Prime Composite (only applicable to QPP-2).

Also
Download data for QPP-2 (2.95MB) and QPP-36 (3MB) up to 100 million - the first ~31,500 prime pairs for each interval.

There is a theorem called Quadratic Reciprocity (QR) that connects perfect squares, modular arithmetic, and primes. Since it involves pairs of primes, one would think there might be an interesting connection with Q-paired primes. An initial comparison of QR primes with QPPs shows that QPPs can have quadratic reciprocity for both, for neither, or for either one of the pair. For primes less than 50 - and looking only at QPP-2 pairs - just one pair is both a QPP and QRP: 13, 29. To make a comparison, compare the following matrices:

QPP Pairs
(Only QPP-2s with both primes under 50 are shown)
QRP Pairs

Q-Paired Primes: The Investigation Continues...

The data's in. (And let me digress for a moment by turning from one passion, number theory, to another, the English language - whether that be the British or American variety. I will never say "the data are in" - this is a linguistic vulgarity. "The statistics are...", yes, but data is a collective noun singular. That it happens to have a virtually unused singular, datum, does not excuse the plural use of data. It is no more a plural noun than "information" is. Do you hear people say the "information are"...? I hope not, but that is how tin-eared "data are" sounds to me. It's not English!* )

As I was saying, the data is in. And the data says there appear to be an infinite number of paired primes for every even - and only even - quadratic interval (QI). If you're not sure what I mean by a paired prime - no, I do not mean twin prime - you must read my previous column about it. I have found that commonly occurring paired primes are not limited to ones separated by the equivalent of exactly 2 QIs (the span of 3 perfect squares), as discussed previously.

There are, in fact, high numbers of paired primes for every even multiple of the quadratic interval: 2, 4, 6, 8, 10, through 100, through 1000, and so on. For simplicity and clarity, I will refer to paired primes of any particular quadratic interval as Q-Paired Primes (QPPs), using a number representing how many intervals the pairs span: QPP-<n>. So QPP-250 would be the designation for paired primes that span 250 quadratic intervals.

You will have a much clearer idea of what I mean by paired if you run the Excel VBA code provided below. However, the screenshot to the right shows some QPP-4s. To take just one pair-prime example from this: 2311 is offset by 7 from perfect square 2304; 2711 is offset by 7 from perfect square 2704.

I hear you say: "So what? Of course you should be able to find primes at the same locations in various quadratic intervals...." But wait and consider the following:

• The locations are not relative; they are absolute. They are ordinal offsets from preceding perfect squares.
• The highest concentrations of paired primes do not occur for the lowest even quadratic interval - namely, 2. In fact, QI-2 ranks 221st out of the first 500 even intervals for paired-prime frequency. (QI-210 ranks highest among the first 500.)
• Since the quadratic interval is growing at f(x) = Θ(x2), the fact that the mean frequency of these pairings diminishes only very gradually as we approach pairs spanning 1,000 quadratic intervals - greater than 1 million at a minimum - is a notable result. Also notable is the steadiness of the frequency fluctuation.

Data for this chart can be downloaded (8KB)

Of the 500 even intervals under 1000, the 10 with the highest paired-prime frequency are as follows:

 Quadratic Intervals Paired Primes (<100,000) 210 3496 30 3380 60 3269 420 3213 120 3172 90 3167 150 3147 630 3137 330 3116 42 3112

The following graph looks at the frequency of paired primes by offset for all numbers under 100 million. The graph shows data for two intervals: QPP-2 and QPP-36 are superimposed, showing the closeness of the distribution for each one.

Data for this chart can be downloaded: QPP-2 (119KB) , QPP-36 (179KB)

There is a striking correspondence in the 10 offsets that yield the highest number of paired primes for each interval:

 For QPP-2 For QPP-36 Quadratic Offset # of Paired Primes Quadratic Offset # of Paired Primes 163 821 163 824 652 784 652 822 1423 726 1423 718 253 703 58 668 58 681 253 668 232 656 2608 661 2608 642 1012 633 928 599 232 624 1087 599 1087 593 1012 594 177 592

Strangely, this is the case even though there is a big difference in the number of offsets producing paired primes for each interval. For QPP-2, there are 12,992 different offsets on which paired primes occur; for QPP-36, there are 19,484 - 67% more paired-prime-producing offsets. The aggregate number of pairs under 100 million is 31,746 for QPP-2 and 31,645 for QPP-36.

Visualizing Paired Primes

I have improved the Excel visualization of paired primes in a number of ways from the original column on biquadratic paired primes:

• It now shows both odd and even offsets.
• It does not repeat numbers.
• It allows you to choose prime-pair quadratic intervals (from 2 to 36) from a list.

(All this functionality is packed into fewer than 100 lines of code - which says more about the effectiveness of VBA for Excel than my programming powers!)

Zoomed out, it looks like this for QPP-10.

All you have to do is to select everything in the text area below and drag or copy it to the Excel Visual Basic Editor, dropping it into the VBAProject ThisWookbook object. Save and exit the file. Then reopen it, and the code will autostart providing that your macro security is not high or very high. When execution stops, change intervals from the selection list in the top left corner and click Run.

Michael M. Ross

Please let me know if you've seen this relationship or an analogous one described elsewhere.