I claim the factorizations below are presently the "beta" factorizations to John Cosgrave's "alpha" factorizations .
For a quick scan by someone with knowledge of a little number theory , perhaps the most interesting parts below are the paragraph which begins with the word "Geometrically" and the section titled "On huge factors of enormous integers" .
On remainders . Dividing a larger integer , called the dividend , by a smaller integer , called the divisor , yields a quotient and a remainder . By convention , the remainder is guaranteed to be an integer between 0 and 1 less than the divisor . ( Otherwise , a lazy wiseacre would dare to tell you that 41 divided by 7 is 0 with a remainder of 41 . ) E.g. , 41 / 7 = 5 , with remainder 6 . On average , the remainder is about 1/2 the divisor . E.g. , if the divisor is 3000000 and the dividend ranges through all 3000000 integers from 3000000000 and 3002999999 , the 3000000 remainders will be each of the integers from 0 to 2999999 , once each , and the average remainder will be 1499999.5 . On primes and factorization . Geometrically , the definition of prime integer is familiar to some 5 and 6yearold children who have played with coins , marbles or toy soldiers , and arranged them in ranks , or equivalently , in rows and columns . For a prime number of objects , rectangularization forces linearization . This is the definition of prime , the most accessible definition of prime . Algebraically , a prime integer is > 1 and has no divisor > 1 , except itself . The Fundamental Theorem of Arithmetic is the foundation of factorization . It asserts that only one breakage of a number into prime factors exists . E.g. , 12 has prime factors 2 , 2 and 3 . Every other set of factors of 12 eventually splits into these , in some order ( together with as many 1's as desired ) . ( People who know a little abstract algebra will recognize that here , number and integer really mean natural or positive integer and that units are disregarded . ) On huge factors of enormous integers . Define np ( n1 ) = n1^n1 + (n1+1)^(n1+1) . For n1 = 1 , 2 and 3 , the first three values are 5 , 31 and 283 . These are the only known primes of this form . Let a = 70 . Let p = np ( a ) / ( 3 * 229 * 1657^2 * 55793 * 54844140931 ) . Let b = p * (p1) + a . p has 107 digits and b has 214 digits . Let c = np ( b ) . Now , p is a prime factor of c . c is far greater than googolplex^googol and , using a small radix , cannot be represented within the known universe . Let a = 802 . Let p = np ( a ) / ( 3 * 7^4 * 13^2 * 29 * 337^2 ) . Let b = p * (p1) + a . p has 2320 digits and b has 4640 digits . Let c = np ( b ) . Now , p is a prime factor of c . Let a = 3376 . Let p = np ( a ) / ( 3 * 7^2 * 13^2 * 19 * 3361 * 41761^2 * 3884884187 ) . p is an 11888digit stronglyprobable prime . p*(p1) + a has 23776 digits . np ( p*(p1) + a ) has more . p divides np ( p*(p1) + a ) . np ( p*(p1) + a ) = 3 * 107 * 14503^2 * 330943^2 * p * c?BIG , where c?BIG is a large , presumably composite , integer . Let a = 4576 . Let p = np ( a ) / ( 3 * 557 * 1609^2 * 4339^2 * 61483 ) . p is an 16733digit stronglyprobable prime . p*(p1) + a has 33466 digits . np ( p*(p1) + a ) has more . p divides np ( p*(p1) + a ) . np ( p*(p1) + a ) = 3 * 7^2 * 193 * 79^2 * 17077 * 43481 * 43577 * 71711 * p * c?BIG Let a = 4576 . Let p = np ( a ) / ( 3 * 557 * 1609^2 * 4339^2 * 61483 ) . p is an 16733digit stronglyprobable prime . Let b = p * (p1) + a . b has 33466 digits . Let c = np ( b ) . c has more . p divides c . c = 3 * 7^2 * 79^2 * 193 * 17077 * 43481 * 43577 * 71711 * p * c?BIG 3 , 7 , 79 , 193 , 17077 , 43481 , 43577 , and 71711 are all prime factors of c . On remainders , again . Asserting that p divides c is a rather strong statement , in a statistical sense . If c were a random enormous integer , dividing it by a 16733digit integer would probably give a 16733 or 16732digit remainder . One of the multitude of "2sigma authorities" on statistics would tell you that it is impossible to find a remainder of less than about 16731 digits . No one would be expected to live long enough to achieve a remainder less than 16000 digits . Yet c divided by p yields a remainder which is not only less than 10000 digits , less than 1000 digits , less than 10 digits , but actually 0 .
Appreciation and links .
It would be ungrateful of me not to mention my debt to powermod =
modexpo = exponentiation under a modulus .
And
GMP,
GMPECM,
NTL,
Primo, and
UBASIC .
In
Cosgrave's
article
look for the phrases "developments in later years" and "light years" .
Some of the larger stronglyprobable primes are accessible under the
name "Walter Nissen" under
Probable Primes .
 Appendix  The largest known such factorizations (4576^4576+4577^4577)/(3*557*1609^2*4339^2*61483),16733 custom code : trial division to 10^8 PrimeForm v. 0.4 Upd.4 : PRP bases 151 , 157 , 163 , 167 custom code : stronglyprobable prime to prime bases 61531061 , 61562411 , 61626029 , 61633909 , 61671833 and 22 others Define np ( n1 ) = n1^n1 + (n1+1)^(n1+1) . Let a = 4576 . Let p = np ( a ) / ( 3 * 557 * 1609^2 * 4339^2 * 61483 ) . p is a 16733digit stronglyprobable prime . p*(p1) + a has 33466 digits . np ( p*(p1) + a ) has more . p divides np ( p*(p1) + a ) . (4223^4223+4224^4224)/(233*185531746148849),15299 custom code : trial division to 10^8 GMP 4.2ECM 6.1 : PRP PrimeForm v. 0.4 Upd.4 : PRP bases 11 , 13 , 233 , 241 , 251 custom code : stronglyprobable prime to prime bases 61962811 , 61963619 , 61977551 , 61979837 and 5 others Define np ( n1 ) = n1^n1 + (n1+1)^(n1+1) . Let a = 4223 . Let p = np ( a ) / ( 233 * 185531746148849 ) . p is an 15299digit stronglyprobable prime . p*(p1) + a has 30597 digits . np ( p*(p1) + a ) has more . p divides np ( p*(p1) + a ) . (3758^3758+3759^3759)/(9067*3706499*250191823789),13417 custom code : trial division to 10^8 GMP 4.2ECM 6.1 : PRP PrimeForm v. 0.4 Upd.4 : PRP bases 17 , 37 , 47 , 67 custom code : stronglyprobable prime to prime bases 61433509 , 61439617 , 61444441 , 61479461 and 3 others Define np ( n1 ) = n1^n1 + (n1+1)^(n1+1) . Let a = 3758 . Let p = np ( a ) / ( 9067 * 3706499 * 250191823789 ) . p is an 13417digit stronglyprobable prime . p*(p1) + a has 26834 digits . np ( p*(p1) + a ) has more . p divides np ( p*(p1) + a ) . (3376^3376+3377^3377)/(3*7^2*13^2*19*3361*41761^2*3884884187),11888 custom code : trial division to 10^8 GMP 4.2ECM 6.1 : PRP PrimeForm v. 0.4 Upd.4 : PRP bases 179 , 191 , 199 , 227 custom code : stronglyprobable prime to prime bases 62063993 , 62064571 , 62065303 , 62071133 and 1 other Define np ( n1 ) = n1^n1 + (n1+1)^(n1+1) . Let a = 3376 . Let p = np ( a ) / ( 3*7^2*13^2*19*3361*41761^2*3884884187 ) . p is an 11888digit stronglyprobable prime . p*(p1) + a has 23776 digits . np ( p*(p1) + a ) has more . p divides np ( p*(p1) + a ) . np ( p*(p1) + a ) = 3 * 107 * 14503^2 * 330943^2 * p?11888 * c?BIG (3294^3294+3295^3295)/12740953,11585 custom code : trial division to 10^8 PrimeForm v. 0.4 Upd.4 : PRP bases 53 , 73 , 79 , 83 , 113 custom code : stronglyprobable prime to prime bases 62076851 , 62081443 , 62081777 , 62083391 and 1 other Define np ( n1 ) = n1^n1 + (n1+1)^(n1+1) . Let a = 3294 . Let p = np ( a ) / 12740953 . p is a 11585digit stronglyprobable prime which divides np ( p*(p1) + a ) . p*(p1) + a has 23169 digits . (3140^3140+3141^3141)/(1571*323819),10976 custom code : trial division to 10^8 PrimeForm v. 0.4 Upd.4 : PRP bases 71 , 113 custom code : stronglyprobable prime to prime bases 62084777 , 62085403 , 62085899 , 62086229 Define np ( n1 ) = n1^n1 + (n1+1)^(n1+1) . Let a = 3140 . Let p = np ( a ) / ( 1571 * 323819 ) . p is a 10976digit probable prime which divides np ( p*(p1) + a ) . p*(p1) + a has 21952 digits .

2000 Mathematics Subject Classification : Primary 11A51 ; Secondary 11A05 , 11A41 , 11A51 , 11N05 , 11N25 , 11Y05 , 11Y11 , 11Y55 .
Walter Nissen
originally posted 20080130
updated 20080616