On Sat, 12 Jul 2014 09:24:26 -0700, djoyce099 wrote:
> Find two semi-primes of the size (e+616) that have a gap < e+157 > > The two semi-primes along with their factors are listed below. > > The two semi-primes difference (gap) between each is e+157. > [snip big semi-primes]
> The record difference (gap) so far is (e+157) between the two semi-primes = > > 1.3233672135852361498935145869900699485939500599345683703707000692473180139638731990175692601515771778417203634445611157686700412066293133925486528829531613736e+157 > > The challenge is to find a difference (gap) of any two semi-primes whose > length = (e+616) for each semi-prime and the gap between the two < than above.
Consider 4 primes p, p+2, q, q+2. Now (p+2)*q - p*(q+2) = 2*(q-p). Because twin primes (like p, p+2 or like q, q+2) are not terrifically infrequent, it isn't difficult to find semiprimes that are close together.
Here is an example to illustrate. In python, import gmpy and then set b = gmpy.next_prime(10**307). Let p = b+44680, q = b+80494. Then p*(q+2) and (p+2)*q are 615-digit semiprimes that differ by 71628.
To get a smaller difference, look for two twin prime pairs closer together, or look for two 4-apart pairs, etc. Eg, let b = gmpy.next_prime(4*10**307), p=b+193626, q=b+202860; then p*(q+4) and (p+4)*q are 616-digit semiprimes that differ by 36936.