Twin Primes with 128 Decimal Digits

[tuckerkao]

2¹⁹¹*3⁹⁰*82589933*40463441610*232792560±1

Click the attached thumbnail to view all the digits in decimal base.

Would like to know why when the number consists 82,589,933 as the factor will result in twin primes at certain power combinations.

[ewmayer]

@tuckertao: It would help if you explained how you chose/found the above pair, and why you think it is somehow special. You have a large-ish smooth part in form of 2^191*3^90, then the largest-known Mersenne prime exponent, but I see nothing special about the remaining multiplier 40463441610*232792560 = 9419588158802421600. If I define a = 2^191*3^90*82589933, set n to your -prime, then successively increment n by a, I can easily find the next-larger pair of this form,

n = 21310867332998259383016283883353020788678719772398547918516685244168507150940923363729663389961621332915816701945558484016693248+-1
 = 2^191*3^90*82589933*9419588158802425428+-1 = 2^193*3^91*82589933*11*29*89*27648398432609+-1.

[tuckerkao]

Originally Posted by ewmayer

@tuckertao: It would help if you explained how you chose/found the above pair, and why you think it is somehow special.]

I chose 2¹⁹¹*3⁹⁰*82589933 because my prime number generator only allowed 128 max digits.

My formula is 2ⁿ*3^m*prime*large abundant number±1

I'm wondering whether it's still possible to find the next larger pair when both n and m are in the millions or larger with a mega-prime possibly the largest known Mersenne Prime.

I go from 2⁰*3^m to 2ⁿ*3⁰, so I have a whole Arc rotation on a given orbit(fix # of numerical digits), doesn't have to be 128 digits, it can be 24 millions+ total.