Revisiting Sums and Differences of Two Pronic Numbers

On Monday the 6th of February 2023, I made a post titled Even Numbers as Sum and Differences of Two Pronic Numbers and concluded it by stating the following:

This post is just meant as an initial investigation into this topic and it would be useful to expand the investigation to include all even numbers in the range up to about 40000.

In this post, I'd like to explore the conjecture that all even numbers can be expressed, at least in one way, as the sum or difference of two pronic numbers. In my original post, I showed that not all even numbers could be expressed as a sum of two pronic numbers nor as a difference of two pronic numbers. However, I never took it further and that's what I'd like to do in this post.

What's prompted this post is one of the properties of the number associated with my diurnal age today: 27332. It is one of those rare numbers that can be expressed as a sum of two pronic numbers in six different ways:

• 27332 is the sum of 27060 and 272
• 27332 is the sum of 26732 and 600
• 27332 is the sum of 25440 and 1892
• 27332 is the sum of 21170 and 6162
• 27332 is the sum of 20022 and 7310
• 27332 is the sum of 17030 and 10302

Let's recall that pronic numbers are of the form \(n \times (n+1) \) for \(n \geq 1\). Up to 40,000, these numbers are:

2, 6, 12, 20, 30, 42, 56, 72, 90, 110, 132, 156, 182, 210, 240, 272, 306, 342, 380, 420, 462, 506, 552, 600, 650, 702, 756, 812, 870, 930, 992, 1056, 1122, 1190, 1260, 1332, 1406, 1482, 1560, 1640, 1722, 1806, 1892, 1980, 2070, 2162, 2256, 2352, 2450, 2550, 2652, 2756, 2862, 2970, 3080, 3192, 3306, 3422, 3540, 3660, 3782, 3906, 4032, 4160, 4290, 4422, 4556, 4692, 4830, 4970, 5112, 5256, 5402, 5550, 5700, 5852, 6006, 6162, 6320, 6480, 6642, 6806, 6972, 7140, 7310, 7482, 7656, 7832, 8010, 8190, 8372, 8556, 8742, 8930, 9120, 9312, 9506, 9702, 9900, 10100, 10302, 10506, 10712, 10920, 11130, 11342, 11556, 11772, 11990, 12210, 12432, 12656, 12882, 13110, 13340, 13572, 13806, 14042, 14280, 14520, 14762, 15006, 15252, 15500, 15750, 16002, 16256, 16512, 16770, 17030, 17292, 17556, 17822, 18090, 18360, 18632, 18906, 19182, 19460, 19740, 20022, 20306, 20592, 20880, 21170, 21462, 21756, 22052, 22350, 22650, 22952, 23256, 23562, 23870, 24180, 24492, 24806, 25122, 25440, 25760, 26082, 26406, 26732, 27060, 27390, 27722, 28056, 28392, 28730, 29070, 29412, 29756, 30102, 30450, 30800, 31152, 31506, 31862, 32220, 32580, 32942, 33306, 33672, 34040, 34410, 34782, 35156, 35532, 35910, 36290, 36672, 37056, 37442, 37830, 38220, 38612, 39006, 39402, 39800

Let's look at numbers in the region of 27332 to get an idea of what's going on (permalink):

• 27334 is the difference of 27390 and 56
• 27334 is the difference of 45156 and 17822
  
  

• 27336 is the sum of 26406 and 930
• 27336 is the difference of 28392 and 1056
• 27336 is the difference of 56406 and 29070
• 27336 is the difference of 86142 and 58806
• 27336 is the difference of 338142 and 310806
• 27336 is the difference of 660156 and 632820
• 27336 is the difference of 2932656 and 2905320
  
  

• 27338 is the difference of 46724060 and 46696722
  
  

• 27340 is the difference of 480942 and 453602
  
  

• 27342 is the difference of 27722 and 380
• 27342 is the sum of 26082 and 1260
• 27342 is the difference of 29412 and 2070
• 27342 is the sum of 22650 and 4692
• 27342 is the difference of 35532 and 8190
• 27342 is the difference of 37442 and 10100
• 27342 is the difference of 61752 and 34410
• 27342 is the difference of 63252 and 35910
• 27342 is the difference of 92112 and 64770
• 27342 is the difference of 120062 and 92720
• 27342 is the difference of 208392 and 181050
• 27342 is the difference of 437582 and 410240
• 27342 is the difference of 590592 and 563250
• 27342 is the difference of 967272 and 939930
• 27342 is the difference of 2321052 and 2293710
• 27342 is the difference of 3827892 and 3800550
  
  

• 27344 is the sum of 23562 and 3782
• 27344 is the sum of 16002 and 11342
• 27344 is the difference of 743906 and 716562
  
  

• 27346 is the difference of 30102 and 2756
• 27346 is the difference of 31506 and 4160
• 27346 is the difference of 400056 and 372710
• 27346 is the difference of 1558752 and 1531406
  
  

• 27348 is the difference of 27390 and 42
• 27348 is the difference of 29070 and 1722
• 27348 is the difference of 80940 and 53592
• 27348 is the difference of 115260 and 87912
• 27348 is the difference of 1312170 and 1284822

At first I couldn't find a sum or difference for 27338 until I extended the list of pronic numbers to 10,000 x 10,001. My initial analysis suggests that it is possible to write any even number as the sum or difference of two pronic numbers. Not surprisingly differences seem to predominate.

At first I thought that 27322 was anomalous in that it is the sum of six different pairs of pronic numbers but no differences. However, extending the list of pronic numbers as before I found that it is also the difference of 11686142 and 11658810. There may well be a formal mathematical proof of my conjecture but my approach so far is purely empirical. Put in formal mathematical terms, I could state my conjecture as follows:

For any even positive integer \(n \gt 2\), there exists pronic numbers \(p \times (p+1) \) and \(q \times (q+1) \) with \(p \gt q\) such that:$$ \begin{align} n &=p \times (p+1) +q \times (q+1) \\ &=p^2+q^2 +p+q \end{align}$$or$$ \begin{align} n &=p \times (p+1) -q \times (q+1) \\ &= p^2-q^2+p-q \\ &= (p+q)(p-q) + p-q \\ &= (p+q)(p-q+1) \end{align}$$

As a final note, let's not forget that every pronic number is twice a triangular number since the formula for a triangular \(T_n\) number is:$$ \begin{align} T_n &= \sum_{k=1}^n k \\ &= 1+2+3 \dots n \\ &=\frac{n \times (n+1)}{2} \\&= \binom{n}{2} \end{align} $$Thus many of the properties of triangular numbers would be shared by pronic numbers.