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 tineared "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 QPaired Primes (QPPs), using a number representing how many intervals the pairs span: QPP<n>. So QPP250 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 QPP4s. To take just one pairprime 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, QI2 ranks 221st out of the first 500 even intervals for pairedprime frequency. (QI210 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 pairedprime 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: QPP2 and QPP36 are superimposed, showing the closeness of the distribution for each one.
Data for this chart can be downloaded: QPP2 (119KB) , QPP36 (179KB)
There is a striking correspondence in the 10 offsets that yield the highest number of paired primes for each interval:
For QPP2

For QPP36

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 QPP2, there are 12,992 different offsets on which paired primes occur; for QPP36, there are 19,484  67% more pairedprimeproducing offsets. The aggregate number of pairs under 100 million is 31,746 for QPP2 and 31,645 for QPP36.
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 primepair 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 QPP10.
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.
