Reversible Twin Prime Composites

What do I mean by a twin prime composite? Well, every twin prime pair has a composite number in between and that's the composite number I'm referring to. For example, 41 and 43 form a twin prime pair and 42 is the composite number in between. What's not so common though is a reversible twin prime composite meaning that when the composite is reversed, it's still wedged between two adjacent primes.

Now 42 fails in this regard because its reversal is 24 and while 23 is prime, 25 is not. However, 60 qualifies because its wedged between the primes 59 and 61 and, when it's reversed to form 6, it is still between two primes (5 and 7). Such numbers form OEIS sequence A103741:

A103741

Non-palindromic composites located between twin primes whose reverses, which are smaller, are also located between twin primes.

Here the condition that the reversed composite must be smaller than the original is imposed. The sequence up to one million is:

60, 240, 270, 600, 810, 822, 2130, 2340, 2802, 8010, 8220, 8430, 8838, 8862, 20550, 22740, 23202, 23370, 23910, 25410, 26880, 27240, 28410, 28572, 28662, 29022, 29760, 80472, 81702, 81930, 81972, 82140, 82530, 83220, 83340, 83640, 85620, 87222, 88470, 203430, 203460, 207240, 208590, 213360, 217200, 218970, 220020, 221070, 224910, 226902, 228300, 230940, 232080, 233160, 233550, 233940, 235440, 238080, 241260, 241512, 243432, 243702, 245130, 245910, 246510, 250050, 250950, 251970, 253680, 255180, 256722, 259122, 259620, 262050, 262152, 263610, 265542, 267390, 267960, 269220, 269430, 270240, 272010, 272760, 278562, 279552, 280410, 281250, 282240, 282390, 286542, 289020, 289140, 292710, 295200, 296730, 296970, 801000, 801420, 802650, 803730, 804282, 806370, 806382, 806790, 808020, 812760, 814062, 815412, 818580, 819618, 820320, 820680, 821208, 822762, 823830, 827130, 829728, 831540, 833712, 834150, 835320, 838092, 840180, 841020, 841080, 846060, 846360, 848790, 848922, 849600, 853902, 856548, 860010, 861492, 862482, 862650, 864300, 864630, 865638, 872658, 872748, 875340, 875418, 875520, 875760, 877110, 878022, 878832, 879168, 879582, 880068, 880248, 880800, 883410, 885552, 887400, 887658, 888060, 888870, 889878, 891000, 893340, 894450, 895650, 895800, 898482, 898662

There are only 168 such numbers in the range up to one million. Notice the big jumps shown in bold. The sequence jumps in a consistent manner:

• 2802 to 8010
• 8862 to 20550
• 29760 to 80472
• 88470 to 203430
• 296970 to 801000

This is best seen graphically. See Figure 1. 

Figure 1: permalink

Why such big gaps? Apart from 60, all numbers start with 2 (and end in 0 or 2) or they start with 8 (and end in 0, 2 or 8). We know that the reverse of the number must be smaller and this means that whatever digit the number starts with, the final digit must be equal to or smaller than it. This immediately rules out a number like 2093. However, why are there no numbers starting with 1, 3, 4, 5, 6, 7 or 9?

The initial composite numbers must be even as they lie between two primes so that rules out numbers whose final digits are 1, 3, 5, 7 or 9. Only composites ending in 0, 2, 4 or 8 are possible. Its reverse must also end in 0, 2, 4 or 8 if it is to lie between twin primes. If a number starts with 2 then it must end in 0 or 2. If a number starts with 4 it must end in 0, 2 or 4. If a number ends in 8 then it must end in 0, 2, 4 or 8. So why are there no numbers starting with or ending in 4? If a number starts with 4, then its reversal will end in a 4 and therefore the number above it (except for 5) cannot be prime because it ends in a 5. 

Figure 2 shows a table of the initial composites and associated primes together with the reversed composites and their associated primes.

Figure 2: permalink