Is this an unknown pattern in prime numbers?
4
$begingroup$
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.
prime-numbers prime-twins
edited Jan 5 at 6:33
Aloizio Macedo♦
23.7k23987
asked Jul 1 '17 at 18:49
Troy WTroy W
474
$endgroup$
|
show 1 more comment
4
$begingroup$
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.
prime-numbers prime-twins
edited Jan 5 at 6:33
Aloizio Macedo♦
23.7k23987
asked Jul 1 '17 at 18:49
Troy WTroy W
474
$endgroup$
5
$begingroup$
I think all you are saying here is that if $p$ and $p+2$ are primes then $pequiv-1pmod 6$. That is easy to prove (with the exception of $p=3$).
$endgroup$
– Lord Shark the Unknown
Jul 1 '17 at 18:54
$begingroup$
Thanks, I see now that it isn't very novel. In radix 6, all the lower twins would automatically have an ending digit of 5 and the higher twins an ending digit of 1.
$endgroup$
– Troy W
Jul 1 '17 at 19:11
$begingroup$
Note that $+1 mod 6$ and $-1 mod 6$ are the accepted notations for what you mean by $6 mod +1$ and $6 mod -1$, respectively.
$endgroup$
– Code-Guru
Jul 1 '17 at 19:23
$begingroup$
@LordSharktheUnknown : That's not all he's saying; there's also a point about multiplication in modular arithmetic.
$endgroup$
– Michael Hardy
Jul 1 '17 at 20:26
2
$begingroup$
@Code-Guru : You can write $+1bmod6$ in MathJax without those huge horizontal spaces, by using bmod instead of mod. The "b" stands for "binary" and it means the spacing should be that which is used for binary operation symbols. That large space is for occasions like this: $$ (52 equiv 64) mod 6, $$ which means $52$ and $64$ both leave the same remainder when divided by $6. qquad$
$endgroup$
– Michael Hardy
Jul 1 '17 at 20:28
|
show 1 more comment
4
4
4
2
$begingroup$
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.
prime-numbers prime-twins
edited Jan 5 at 6:33
Aloizio Macedo♦
23.7k23987
asked Jul 1 '17 at 18:49
Troy WTroy W
474
$endgroup$
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.
prime-numbers prime-twins
prime-numbers prime-twins
edited Jan 5 at 6:33
Aloizio Macedo♦
23.7k23987
asked Jul 1 '17 at 18:49
Troy WTroy W
474
edited Jan 5 at 6:33
Aloizio Macedo♦
23.7k23987
asked Jul 1 '17 at 18:49
Troy WTroy W
474
edited Jan 5 at 6:33
Aloizio Macedo♦
23.7k23987
edited Jan 5 at 6:33
Aloizio Macedo♦
23.7k23987
edited Jan 5 at 6:33
Aloizio Macedo♦
23.7k23987
23.7k23987
asked Jul 1 '17 at 18:49
Troy WTroy W
474
asked Jul 1 '17 at 18:49
Troy WTroy W
474
asked Jul 1 '17 at 18:49
Troy WTroy W
474
474
5
$begingroup$
I think all you are saying here is that if $p$ and $p+2$ are primes then $pequiv-1pmod 6$. That is easy to prove (with the exception of $p=3$).
$endgroup$
– Lord Shark the Unknown
Jul 1 '17 at 18:54
$begingroup$
Thanks, I see now that it isn't very novel. In radix 6, all the lower twins would automatically have an ending digit of 5 and the higher twins an ending digit of 1.
$endgroup$
– Troy W
Jul 1 '17 at 19:11
$begingroup$
Note that $+1 mod 6$ and $-1 mod 6$ are the accepted notations for what you mean by $6 mod +1$ and $6 mod -1$, respectively.
$endgroup$
– Code-Guru
Jul 1 '17 at 19:23
$begingroup$
@LordSharktheUnknown : That's not all he's saying; there's also a point about multiplication in modular arithmetic.
$endgroup$
– Michael Hardy
Jul 1 '17 at 20:26
2
$begingroup$
@Code-Guru : You can write $+1bmod6$ in MathJax without those huge horizontal spaces, by using bmod instead of mod. The "b" stands for "binary" and it means the spacing should be that which is used for binary operation symbols. That large space is for occasions like this: $$ (52 equiv 64) mod 6, $$ which means $52$ and $64$ both leave the same remainder when divided by $6. qquad$
$endgroup$
– Michael Hardy
Jul 1 '17 at 20:28
|
show 1 more comment
5
$begingroup$
I think all you are saying here is that if $p$ and $p+2$ are primes then $pequiv-1pmod 6$. That is easy to prove (with the exception of $p=3$).
$endgroup$
– Lord Shark the Unknown
Jul 1 '17 at 18:54
$begingroup$
Thanks, I see now that it isn't very novel. In radix 6, all the lower twins would automatically have an ending digit of 5 and the higher twins an ending digit of 1.
$endgroup$
– Troy W
Jul 1 '17 at 19:11
$begingroup$
Note that $+1 mod 6$ and $-1 mod 6$ are the accepted notations for what you mean by $6 mod +1$ and $6 mod -1$, respectively.
$endgroup$
– Code-Guru
Jul 1 '17 at 19:23
$begingroup$
@LordSharktheUnknown : That's not all he's saying; there's also a point about multiplication in modular arithmetic.
$endgroup$
– Michael Hardy
Jul 1 '17 at 20:26
2
$begingroup$
@Code-Guru : You can write $+1bmod6$ in MathJax without those huge horizontal spaces, by using bmod instead of mod. The "b" stands for "binary" and it means the spacing should be that which is used for binary operation symbols. That large space is for occasions like this: $$ (52 equiv 64) mod 6, $$ which means $52$ and $64$ both leave the same remainder when divided by $6. qquad$
$endgroup$
– Michael Hardy
Jul 1 '17 at 20:28
5
5
$begingroup$
I think all you are saying here is that if $p$ and $p+2$ are primes then $pequiv-1pmod 6$. That is easy to prove (with the exception of $p=3$).
$endgroup$
– Lord Shark the Unknown
Jul 1 '17 at 18:54
$begingroup$
I think all you are saying here is that if $p$ and $p+2$ are primes then $pequiv-1pmod 6$. That is easy to prove (with the exception of $p=3$).
$endgroup$
– Lord Shark the Unknown
Jul 1 '17 at 18:54
$begingroup$
Thanks, I see now that it isn't very novel. In radix 6, all the lower twins would automatically have an ending digit of 5 and the higher twins an ending digit of 1.
$endgroup$
– Troy W
Jul 1 '17 at 19:11
$begingroup$
Thanks, I see now that it isn't very novel. In radix 6, all the lower twins would automatically have an ending digit of 5 and the higher twins an ending digit of 1.
$endgroup$
– Troy W
Jul 1 '17 at 19:11
$begingroup$
Note that $+1 mod 6$ and $-1 mod 6$ are the accepted notations for what you mean by $6 mod +1$ and $6 mod -1$, respectively.
$endgroup$
– Code-Guru
Jul 1 '17 at 19:23
$begingroup$
Note that $+1 mod 6$ and $-1 mod 6$ are the accepted notations for what you mean by $6 mod +1$ and $6 mod -1$, respectively.
$endgroup$
– Code-Guru
Jul 1 '17 at 19:23
$begingroup$
@LordSharktheUnknown : That's not all he's saying; there's also a point about multiplication in modular arithmetic.
$endgroup$
– Michael Hardy
Jul 1 '17 at 20:26
$begingroup$
@LordSharktheUnknown : That's not all he's saying; there's also a point about multiplication in modular arithmetic.
$endgroup$
– Michael Hardy
Jul 1 '17 at 20:26
2
2
$begingroup$
@Code-Guru : You can write $+1bmod6$ in MathJax without those huge horizontal spaces, by using bmod instead of mod. The "b" stands for "binary" and it means the spacing should be that which is used for binary operation symbols. That large space is for occasions like this: $$ (52 equiv 64) mod 6, $$ which means $52$ and $64$ both leave the same remainder when divided by $6. qquad$
$endgroup$
– Michael Hardy
Jul 1 '17 at 20:28
$begingroup$
@Code-Guru : You can write $+1bmod6$ in MathJax without those huge horizontal spaces, by using bmod instead of mod. The "b" stands for "binary" and it means the spacing should be that which is used for binary operation symbols. That large space is for occasions like this: $$ (52 equiv 64) mod 6, $$ which means $52$ and $64$ both leave the same remainder when divided by $6. qquad$
$endgroup$
– Michael Hardy
Jul 1 '17 at 20:28
|
show 1 more comment
1 Answer
1
active
oldest
votes
4
$begingroup$
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 $aequiv Abmod 6$ and $bequiv Bbmod 6$ then $abequiv ABbmod6$. Proving that takes a bit of elementary algebra. Applying it here we have $7equiv 1;$ therefore $7times7equiv 1times 1,$ etc.
The fact that twin primes are always of the form $6npm1,$ plus the facts above lead to the conclusion that the pattern you've identified will persist.
answered Jul 1 '17 at 19:17
Michael HardyMichael Hardy
1
$endgroup$
add a comment |
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$begingroup$
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 $aequiv Abmod 6$ and $bequiv Bbmod 6$ then $abequiv ABbmod6$. Proving that takes a bit of elementary algebra. Applying it here we have $7equiv 1;$ therefore $7times7equiv 1times 1,$ etc.
The fact that twin primes are always of the form $6npm1,$ plus the facts above lead to the conclusion that the pattern you've identified will persist.
answered Jul 1 '17 at 19:17
Michael HardyMichael Hardy
1
$endgroup$
add a comment |
4
$begingroup$
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 $aequiv Abmod 6$ and $bequiv Bbmod 6$ then $abequiv ABbmod6$. Proving that takes a bit of elementary algebra. Applying it here we have $7equiv 1;$ therefore $7times7equiv 1times 1,$ etc.
The fact that twin primes are always of the form $6npm1,$ plus the facts above lead to the conclusion that the pattern you've identified will persist.
answered Jul 1 '17 at 19:17
Michael HardyMichael Hardy
1
$endgroup$
add a comment |
4
4
4
$begingroup$
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 $aequiv Abmod 6$ and $bequiv Bbmod 6$ then $abequiv ABbmod6$. Proving that takes a bit of elementary algebra. Applying it here we have $7equiv 1;$ therefore $7times7equiv 1times 1,$ etc.
The fact that twin primes are always of the form $6npm1,$ plus the facts above lead to the conclusion that the pattern you've identified will persist.
answered Jul 1 '17 at 19:17
Michael HardyMichael Hardy
1
$endgroup$
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 $aequiv Abmod 6$ and $bequiv Bbmod 6$ then $abequiv ABbmod6$. Proving that takes a bit of elementary algebra. Applying it here we have $7equiv 1;$ therefore $7times7equiv 1times 1,$ etc.
The fact that twin primes are always of the form $6npm1,$ plus the facts above lead to the conclusion that the pattern you've identified will persist.
answered Jul 1 '17 at 19:17
Michael HardyMichael Hardy
1
answered Jul 1 '17 at 19:17
Michael HardyMichael Hardy
1
answered Jul 1 '17 at 19:17
Michael HardyMichael Hardy
1
answered Jul 1 '17 at 19:17
Michael HardyMichael Hardy
1
1
add a comment |
add a comment |
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I think all you are saying here is that if $p$ and $p+2$ are primes then $pequiv-1pmod 6$. That is easy to prove (with the exception of $p=3$).
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– Lord Shark the Unknown
Jul 1 '17 at 18:54
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Thanks, I see now that it isn't very novel. In radix 6, all the lower twins would automatically have an ending digit of 5 and the higher twins an ending digit of 1.
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– Troy W
Jul 1 '17 at 19:11
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Note that $+1 mod 6$ and $-1 mod 6$ are the accepted notations for what you mean by $6 mod +1$ and $6 mod -1$, respectively.
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– Code-Guru
Jul 1 '17 at 19:23
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@LordSharktheUnknown : That's not all he's saying; there's also a point about multiplication in modular arithmetic.
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– Michael Hardy
Jul 1 '17 at 20:26
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@Code-Guru : You can write $+1bmod6$ in MathJax without those huge horizontal spaces, by using bmod instead of mod. The "b" stands for "binary" and it means the spacing should be that which is used for binary operation symbols. That large space is for occasions like this: $$ (52 equiv 64) mod 6, $$ which means $52$ and $64$ both leave the same remainder when divided by $6. qquad$
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– Michael Hardy
Jul 1 '17 at 20:28