NUMBER-FRACTION, SUPER-COMPOSITES AND FACTORIAL PRIMES

We re-look at super-composites and their properties and use this information to define a new sub-set of primes defined as N!1. As a starting point we note that the number-fraction is defined as-

[sigma(N )  N 1] f (N )  N

This point function vanishes whenever N is a prime, has a value less than one for semi-primes and most composites, but takes on values greater than unity for super-primes.These super-primes typically will have many more divisors than its neighbors as shown in the following example for N=21600-

Such super-primes tower above neighbors as the graph indicates. Also they sometimes will have prime values located at N+1 or N-1. When such super- composites are expressed as product of primes taken to specified powers they typically will have the form-

N  2a  3b 5c  7d ... where a  b  c  d...

, with the highest power reserved for the first prime 2. For the above case of N=21600 we have-

ifactor(N)=253352

So one can easily find super-primes with f(N)>1 by just writing down the first few primes taken to specified descending powers. Consider the number-

N  24  32 5 7  5040

It has f(N)= 2.8378998 and so is a super-composite. The following graph confirms this point-

It also shows a prime at N-1 and a semi-prime at N+1.

When looking at the numbers 24=4!=233 and 120=5!=2335 we notice that they have the form required for super-composites. This is confirmed by noting that f(24)= 1.458333333…and f(120)= 1.991666667 . Furthermore we note that 24-1=23 is a prime but neither 119= 717 or 121=1111 are. This suggests to us that there exist a heretofore unrecognized sub-set of primes defined by N!1.To get the value of these we start with the following table-

N N! Product Breakup f(N) 2 2 2 0

3 6 23 0.8333333335 4 24 233 1.458333333 5 120 2335 1.991666667 6 720 24325 2.356944445 7 5040 243257 2.837896825 8 40320 273257 2.946403770 9 362880 273457 3.081346450 10 3628800 2834527 3.225663304 11 39916800 283452711 3.609814790 12 479001600 2103552711 3.629298539 13 6227020800 210355271113 3.985398429 14 87178291200 2113552721113 4.075662880 15 1307674368000 2113653721113 4.113087121 16 20922789888000 2153653721113 4.114257700 17 355687428096000 215365372111317 4.415096387 18 6402373705728000 216385372111317 4.417339650

Here all factorials starting with 4!=24 are super-composites with the value of f(N) increasing slowly toward infinity as N gets large. It is easy to construct this table since the product form for N! just needs to be multiplied by the product form of N+1. Thus the product form of 8! will be (243257)(23)= 273257. Note also that all factorials after 4 end in zero with the number of zeros in the ending increasing with N!.

Next let us get the primes in the immediate vicinity of N!. We call this special subset of primes the factorial primes. They are found at N!1. To find them we go to the following computer program-

for N from 2 to B do {N,N!,isprime(N+1),isprime(N-1)}od;

We ran the program in chunks of approximately 50 so that the first calculation went from 2 to 50. The second from 50 through 100 and so on. The calculations where taken up through B=427, the run times becoming longer as N increased. Here are the results for the first 27 factorial primes-

3!-1, 3!+1, 4!-1, 6!-1, 7!-1, 11!+1, 12!-1, 14!-1, 27!+1, 30!-1, 32!-1, 33!-1, 37!+1, 38!-1, 41!+1, 73!+1, 77!+1, 94!-1, 116!+1, 154!+1, 166!-1, 320!+1, 324!-1, 340!+1, 379!-1, 399!+1, 427!+1

A graph of the super-composite 12!=479001600 , also showing the prime 12!-1, follows-

As N gets large, the N! will become huge. For N=427! We get-

427!= 290634717696073484110291540071593168467673838691755129360254894001708915 156302009607579314550754490979941932585979532362609329479760129536206622 244743895266703131541072488397502825504504125096700302164981563491767272 351215204631732981320437073122067583998534799549602620384654559505624578 854698733575026020835828454629114747329325251579113972247313188408636049 674932065755112783978528908261023662539783841799404414005206452815755145 762057823239143599179132575846354941784194937658945303199938869683107887 341107459990963377836951125261421511684590201163748678649634403303021152 060749847407434871658310342772480238778960424520645773241762074192212826 904696038697541329479463378151296849064283833380595279650338771184440530 651498776584346035506476674424138813956488251447966048576070679639809134 312397346773437181822436621881409132088524800000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000 0000

This represents a 940 digit long super-composite with f(N)= 9.844249994. Its ifactor reads-

2421  3210  5105  770  1141  1334 1726  1923  2318  2914……

with many more product terms with descending powers present in the expansion ending in 269271)277. Note that the highest powers are

associated with the lowest primes 2,3,5,7 as expected of all super- composites. The corresponding prime is 427!+1 and so also has 940 digits. The largest known Mersenne Prime as of January of this year is 277232917-1. It has a length of 23249425 digits. It would seem that one could find, without too much extra work, a factorial prime larger than this. This would then make such a factorial prime the largest known prime.

U.H.Kurzweg Nov. 22, 2018 Thanksgiving