I am trying to figure out if the pattern I've found concerning twin primes is a known pattern or not. It turns out that with every set of twin primes, if the higher of the two numbers is converted to radix 7, and then the individual digits of the number are added together and continually added together until a 1 or 2 digit number is leftover, the number is always equal to 6 mod +1.
The lower of the two numbers is always 6 mod -1 with same calculation.
Examples:
Lower twin (radix 10)/ Lower twin (radix 7)/ [sum of digits to 2 digits]/ MOD 6
<ul>59 / 113 / 5 / 5</ul>
<ul>71 / 131 / 5 / 5</ul>
<ul>101 / 203 / 5 / 5</ul>
<ul>107 / 212 / 5 / 5</ul>
<ul>137 / 254 / 11 / 5</ul>
<ul>149 / 302 / 5 / 5</ul>
<ul>179 / 344 / 11 / 5</ul>
<ul>191 / 362 / 11 / 5</ul>
<ul>197 / 401 / 5 / 5</ul>
<ul>227 / 443 / 11 / 5</ul>
<ul>239 / 461 / 11 / 5</ul>
<ul>269 / 533 / 11 / 5</ul>
<ul>281 / 551 / 11 / 5</ul>
<ul>311 / 623 / 11 / 5</ul>
<ul>347 / 1004 / 5 / 5</ul>
<ul>419 / 1136 / 11 / 5</ul>
<ul>431 / 1154 / 11 / 5</ul>
<ul>461 / 1226 / 11 / 5</ul>
<ul>521 / 1343 / 11 / 5</ul>
<ul>569 / 1442 / 11 / 5</ul>
<ul>599 / 1514 / 11 / 5</ul>
<ul>617 / 1541 / 11 / 5</ul>
<ul>641 / 1604 / 11 / 5</ul>
<ul>659 / 1631 / 11 / 5</ul>
<ul>809 / 2234 / 11 / 5</ul>
<ul>821 / 2252 / 11 / 5</ul>
<ul>827 / 2261 / 11 / 5</ul>
<ul>857 / 2333 / 11 / 5</ul>
<ul>881 / 2366 / 17 / 5</ul>
<ul>1019 / 2654 / 17 / 5</ul>
<ul>1031 / 3002 / 5 / 5</ul>
<ul>1049 / 3026 / 11 / 5</ul>
<ul>1061 / 3044 / 11 / 5</ul>
<ul>1091 / 3116 / 11 / 5</ul>
<ul>1151 / 3233 / 11 / 5</ul>
<ul>1229 / 3404 / 11 / 5</ul>
<ul>1277 / 3503 / 11 / 5</ul>
<ul>1289 / 3521 / 11 / 5</ul>
<ul>1301 / 3536 / 17 / 5</ul>
<ul>1319 / 3563 / 17 / 5</ul>
<ul>1427 / 4106 / 11 / 5</ul>
<ul>1451 / 4142 / 11 / 5</ul>
<ul>1481 / 4214 / 11 / 5</ul>
<ul>1487 / 4223 / 11 / 5</ul>
<ul>1607 / 4454 / 17 / 5</ul>
<ul>1619 / 4502 / 11 / 5</ul>
<ul>963426767 / 32605664252 / 41 / 5</ul>
<ul>963427259 / 32605665554 / 47 / 5</ul>
<ul>963427301 / 32605665644 / 47 / 5</ul>
<ul>963427559 / 32605666463 / 47 / 5</ul>
<ul>963427919 / 32606000516 / 29 / 5</ul>
<ul>963428021 / 32606001023 / 23 / 5</ul>
<ul>963428099 / 32606001164 / 29 / 5</ul>
<ul>963428561 / 32606002424 / 29 / 5</ul>
<ul>963428861 / 32606003333 / 29 / 5</ul>
<ul>963428957 / 32606003531 / 29 / 5</ul>
<ul>963429167 / 32606004251 / 29 / 5</ul>
<ul>963430019 / 32606006606 / 35 / 5</ul>
<ul>963430079 / 32606010023 / 23 / 5</ul>
<ul>963430289 / 32606010443 / 29 / 5</ul>
<ul>963431177 / 32606013152 / 29 / 5</ul>
<ul>963431321 / 32606013446 / 35 / 5</ul>
<ul>963431477 / 32606014061 / 29 / 5</ul>
<ul>963431717 / 32606014553 / 35 / 5</ul>
<ul>963432131 / 32606016014 / 29 / 5</ul>
<ul>963432917 / 32606021216 / 29 / 5</ul>
<ul>963432989 / 32606021351 / 29 / 5</ul>
<ul>963433319 / 32606022332 / 29 / 5</ul>
<ul>963433439 / 32606022563 / 35 / 5</ul>
<ul>963433697 / 32606023412 / 29 / 5</ul>
<ul>963434411 / 32606025452 / 35 / 5</ul>
<ul>963434579 / 32606026112 / 29 / 5</ul>
<ul>963434609 / 32606026154 / 35 / 5</ul>
<ul>963434891 / 32606030036 / 29 / 5</ul>
<ul>963435227 / 32606031026 / 29 / 5</ul>
<ul>963435491 / 32606031554 / 35 / 5</ul>
<ul>963436037 / 32606033264 / 35 / 5</ul>
<ul>963436601 / 32606035031 / 29 / 5</ul>
<ul>963437261 / 32606036663 / 41 / 5</ul>
<ul>963437399 / 32606040251 / 29 / 5</ul>
<ul>963437927 / 32606041634 / 35 / 5</ul>
<ul>963437939 / 32606041652 / 35 / 5</ul>
<ul>963438017 / 32606042123 / 29 / 5</ul>
<ul>963438041 / 32606042156 / 35 / 5</ul>
Higher twin (radix 10)/ Higher twin (radix 7)/ [sum of digits to 2 digits]/ MOD 6
<ul>571 / 1444 / 13 / 1</ul>
<ul>601 / 1516 / 13 / 1</ul>
<ul>619 / 1543 / 13 / 1</ul>
<ul>643 / 1606 / 13 / 1</ul>
<ul>661 / 1633 / 13 / 1</ul>
<ul>811 / 2236 / 13 / 1</ul>
<ul>823 / 2254 / 13 / 1</ul>
<ul>829 / 2263 / 13 / 1</ul>
<ul>859 / 2335 / 13 / 1</ul>
<ul>883 / 2401 / 7 / 1</ul>
<ul>1021 / 2656 / 19 / 1</ul>
<ul>1033 / 3004 / 7 / 1</ul>
<ul>1051 / 3031 / 7 / 1</ul>
<ul>1063 / 3046 / 13 / 1</ul>
<ul>1093 / 3121 / 7 / 1</ul>
<ul>1153 / 3235 / 13 / 1</ul>
<ul>1231 / 3406 / 13 / 1</ul>
<ul>1279 / 3505 / 13 / 1</ul>
<ul>1291 / 3523 / 13 / 1</ul>
<ul>1303 / 3541 / 13 / 1</ul>
<ul>1321 / 3565 / 19 / 1</ul>
<ul>1429 / 4111 / 7 / 1</ul>
<ul>1453 / 4144 / 13 / 1</ul>
<ul>1483 / 4216 / 13 / 1</ul>
<ul>961750903 / 32555514331 / 37 / 1</ul>
<ul>961751209 / 32555515246 / 43 / 1</ul>
<ul>961752301 / 32555521366 / 43 / 1</ul>
<ul>961752349 / 32555521465 / 43 / 1</ul>
<ul>961752553 / 32555522206 / 37 / 1</ul>
<ul>961753789 / 32555525623 / 43 / 1</ul>
<ul>961753831 / 32555526013 / 37 / 1</ul>
<ul>961754011 / 32555526361 / 43 / 1</ul>
<ul>961754071 / 32555526505 / 43 / 1</ul>
<ul>961754461 / 32555530603 / 37 / 1</ul>
<ul>961755019 / 32555532331 / 37 / 1</ul>
<ul>961757059 / 32555541304 / 37 / 1</ul>
<ul>961757311 / 32555542114 / 37 / 1</ul>
<ul>961757431 / 32555542345 / 43 / 1</ul>
<ul>961757683 / 32555543155 / 43 / 1</ul>
<ul>961758673 / 32555546101 / 37 / 1</ul>
<ul>961759111 / 32555550265 / 43 / 1</ul>
<ul>961759483 / 32555551336 / 43 / 1</ul>
<ul>961759831 / 32555552344 / 43 / 1</ul>
<ul>961759861 / 32555552416 / 43 / 1</ul>
<ul>961760119 / 32555553235 / 43 / 1</ul>
<ul>961760719 / 32555555053 / 43 / 1</ul>
<ul>961761013 / 32555555653 / 49 / 1</ul>
<ul>961761139 / 32555556223 / 43 / 1</ul>
<ul>961761343 / 32555556634 / 49 / 1</ul>
<ul>961761403 / 32555560051 / 37 / 1</ul>
<ul>961761571 / 32555560411 / 37 / 1</ul>
<ul>961762033 / 32555561641 / 43 / 1</ul>
<ul>961762591 / 32555563366 / 49 / 1</ul>
I have other questions related to prime numbers but first want to see how valid or known this part is before I continue. I am not a mathematician.
\begin{align} d_0 + 7d_1 + 7^2 d_2 + 7^3 d_3 + \cdots & \equiv d_0 + 1d_1 + 1^2 d_2 + 1^3 d_3+\cdots & &\mod 6 \\[10pt] & \equiv d_0 + d_1 + d_2 + d_3 + \cdots & & \mod 6 \end{align}
What is at work here is something that says if $a\equiv A\bmod 6$ and $b\equiv B\bmod 6$ then $ab\equiv AB\bmod6$. Proving that takes a bit of elementary algebra. Applying it here we have $7\equiv 1;$ therefore $7\times7\equiv 1\times 1,$ etc.
The fact that twin primes are always of the form $6n\pm1,$ plus the facts above lead to the conclusion that the pattern you've identified will persist.