It was a form of Goldbach's conjecture that got me thinking about an equivalent involving pronic numbers. The conjecture is that every even number can be written as a sum of two prime numbers. Pronic numbers, that is numbers of the form \(n \times (n-1) \), are always even and so the sum of two pronic numbers must be even as well.
Can every even number be written as a sum of two not necessarily distinct pronic numbers. Let's consider the numbers from 1 to 100. The results are as follows:
- 4 = 2 + 2
- 8 = 2 + 6
- 12 = 6 + 6
- 14 = 2 + 12
- 18 = 6 + 12
- 22 = 2 + 20
- 24 = 12 + 12
- 26 = 6 + 20
- 32 = 2 + 30
- 32 = 12 + 20
- 36 = 6 + 30
- 40 = 20 + 20
- 42 = 12 + 30
- 44 = 2 + 42
- 48 = 6 + 42
- 50 = 20 + 30
- 54 = 12 + 42
- 58 = 2 + 56
- 60 = 30 + 30
- 62 = 6 + 56
- 62 = 20 + 42
- 68 = 12 + 56
- 72 = 30 + 42
- 74 = 2 + 72
- 76 = 20 + 56
- 78 = 6 + 72
- 84 = 12 + 72
- 84 = 42 + 42
- 86 = 30 + 56
- 92 = 2 + 90
- 92 = 20 + 72
- 96 = 6 + 90
- 98 = 42 + 56
As can be seen, not all even numbers between 4 and 100 can be written as a sum of two pronic numbers. We are missing 6, 10, 16, 20, 28, 30, 34, 38, 46, 52, 56, 64, 70, 80, 82, 88, 90, 94 and 100. Some numbers however, can be written as a sum of two pronic numbers in more than one way. These numbers are 32, 62, 84 and 92. Each can be written in two ways.
Let's take another range of one hundred numbers, this time from 26900 to 27000. The results are as follows:
- 26906 = 4556 + 22350
- 26912 = 506 + 26406
- 26912 = 6320 + 20592
- 26912 = 13340 + 13572
- 26914 = 182 + 26732
- 26916 = 3660 + 23256
- 26916 = 8010 + 18906
- 26916 = 8556 + 18360
- 26916 = 13110 + 13806
- 26922 = 1482 + 25440
- 26922 = 10920 + 16002
- 26924 = 12882 + 14042
- 26928 = 1806 + 25122
- 26930 = 9900 + 17030
- 26936 = 2756 + 24180
- 26936 = 12656 + 14280
- 26940 = 4290 + 22650
- 26942 = 210 + 26732
- 26942 = 2450 + 24492
- 26942 = 7482 + 19460
- 26942 = 9120 + 17822
- 26948 = 6642 + 20306
- 26950 = 1190 + 25760
- 26950 = 3080 + 23870
- 26952 = 870 + 26082
- 26952 = 12432 + 14520
- 26958 = 552 + 26406
- 26968 = 2162 + 24806
- 26968 = 10712 + 16256
- 26972 = 240 + 26732
- 26972 = 12210 + 14762
- 26984 = 3422 + 23562
- 26984 = 4032 + 22952
- 26994 = 6972 + 20022
- 26994 = 9702 + 17292
- 26996 = 11990 + 15006
- 27000 = 1560 + 25440
There are 21 numbers in the range of 51 even numbers that can be expressed as a sum of two pronic numbers. This is a percentage of about 42%. These are (permalink):
26906, 26912, 26912, 26912, 26914, 26916, 26916, 26916, 26916, 26922, 26922, 26924, 26928, 26930, 26936, 26936, 26940, 26942, 26942, 26942, 26942, 26948, 26950, 26950, 26952, 26952, 26958, 26968, 26968, 26972, 26972, 26984, 26984, 26994, 26994, 26996, 27000
Some numbers can be expressed as a sum in four ways. Here is the count of ways with four ways numbers marked in bold):
(26906, 1), (26912, 3), (26914, 1), (26916, 4), (26922, 2), (26924, 1), (26928, 1), (26930, 1), (26936, 2), (26940, 1), (26942, 4), (26948, 1), (26950, 2), (26952, 2), (26958, 1), (26968, 2), (26972, 2), (26984, 2), (26994, 2), (26996, 1), (27000, 1)
Thus is the range selected there are only two numbers that meet the four ways criterion and these are 26916 and 26942. This is a percentage of about 4%. What becomes of interest then is what numbers in a given range can be expressed as a sum of two pronic numbers in exactly four ways. Let's consider a range of two thousand, from 26000 to 28000. There are only 33 such numbers, representing about 3.3% of the range, and they are (permalink):
26032, 26052, 26102, 26214, 26222, 26292, 26312, 26342, 26382, 26432, 26630, 26682, 26752, 26868, 26916, 26942, 27042, 27062, 27072, 27102, 27176, 27192, 27242, 27402, 27480, 27522, 27572, 27602, 27752, 27852, 27854, 27922, 27990
Some numbers can be written is five different ways. There are two numbers in the range between 26000 and 28000 and they are 26732 and 27812. Other numbers can be written as a sum of pronic numbers in six different ways. These numbers are 26462, 26562, 27162, 27332 and 27462. Below is shown the sums for 27462:
- 27462 = 72 + 27390
- 27462 = 1056 + 26406
- 27462 = 2970 + 24492
- 27462 = 5112 + 22350
- 27462 = 8556 + 18906
- 27462 = 12210 + 15252
There are no numbers in the range that can be expressed as a sum in seven ways. Notice that throughout this investigation, we imposed the condition that the two pronic numbers need not be distinct. If we imposed the condition that they must be distinct, then there would be slightly fewer numbers that satisfy.
It's also possible to consider numbers that are a difference of two pronic numbers. For example in the range between 26900 and 27000, there are 35 numbers out the 51 even numbers that can expressed as differences, many in multiple ways. However, there are 16 that cannot and these are (permalink):
26902, 26914, 26916, 26926, 26930, 26932, 26938, 26948, 26954, 26958, 26968, 26974, 26982, 26984, 26990, 26998
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.
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