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• I m referring to composite numbers ending in 1,3,7 or 9. Noting that 403=13*31, 407=11*37, and also 1003=17*59, 1007=19*53; in both cases there are adjacent
Message 1 of 2 , Jun 14, 2012
I'm referring to composite numbers ending in 1,3,7 or 9.

Noting that 403=13*31, 407=11*37, and also 1003=17*59, 1007=19*53; in
both cases there are adjacent composite numbers (as above) with pairs of
adjacent prime factors (11,13), (31,37), (17,19) and (53,59).

I have no idea of how infrequently such cases occur - does anybody have

[Non-text portions of this message have been removed]
• ... I found these other cases : [[403, 407, [13, 31], [11, 37]]] [[1003, 1007, [17, 59], [19, 53]]] [[110203, 110207, [193, 571], [191, 577]]] [[118003,
Message 2 of 2 , Jun 15, 2012
>>> > When you say "such cases", do you include the condition that they are
>>> > semiprimes, or only that they have two pairs of adjacent prime factors?
>> Yes, they must also be semiprimes.

I found these other cases :

[[403, 407, [13, 31], [11, 37]]]
[[1003, 1007, [17, 59], [19, 53]]]
[[110203, 110207, [193, 571], [191, 577]]]
[[118003, 118007, [197, 599], [199, 593]]]
[[418307, 418309, [557, 751], [563, 743]]]
[[429491, 429493, [311, 1381], [307, 1399]]]
[[439097, 439099, [577, 761], [571, 769]]]
[[559003, 559007, [433, 1291], [431, 1297]]]
[[1239869, 1239871, [907, 1367], [911, 1361]]]
[[1887239, 1887241, [1249, 1511], [1259, 1499]]]
[[2481463, 2481467, [1217, 2039], [1223, 2029]]]
[[2502977, 2502979, [1367, 1831], [1373, 1823]]]
[[3381403, 3381407, [1063, 3181], [1061, 3187]]]
[[3693419, 3693421, [1567, 2357], [1571, 2351]]]
[[5646257, 5646259, [1103, 5119], [1097, 5147]]]
[[6120403, 6120407, [1427, 4289], [1429, 4283]]]
[[6586003, 6586007, [1483, 4441], [1481, 4447]]]
[[6954767, 6954769, [2287, 3041], [2281, 3049]]]
[[7042661, 7042663, [967, 7283], [971, 7253]]]
[[8350003, 8350007, [1667, 5009], [1669, 5003]]]
[[11305093, 11305097, [1759, 6427], [1753, 6449]]]
[[13083403, 13083407, [2087, 6269], [2089, 6263]]]
[[13760203, 13760207, [2143, 6421], [2141, 6427]]]
[[17297519, 17297521, [1699, 10181], [1697, 10193]]]
[[21159419, 21159421, [1879, 11261], [1877, 11273]]]
...

Maximilian

PS:
{e=[1,3,7,9,11]; forstep(a=10,9e9,10, for( i=2,#e,
bigomega(a+e[i])==2 | (i++ & next);
bigomega(a+e[i-1]) ==2 | next; f=factor(a+e[i-1])[,1];
nextprime(f[1]+1)*precprime(vecmax(f)-1)==a+e[i]
| precprime(f[1]-1)*nextprime(vecmax(f)+1)==a+e[i]
| next; print([a+e[i-1],a+e[i],f~,factor(a+e[i])[,1]~]", ")))}
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