Concerning types of square-free numbers.

Question

Call a square-free number a 3-prime if it is the product of three primes. Similarly for 2-primes, 4-primes , 5-primes, etc. Are there two consecutive 3-primes with no 2-prime between them?Are there infinitely many?

Answer 1

I hope I got the programming right: I get $679$ pairs less than $10000$, of which the first few are $$ \eqalign{ [102,105], &[170,174], &[230,231], &[238,246], &[255,258], &[282,285], &[285,286],\cr [366,370], &[399,402], &[429,430], &[430,434], &[434,435], &[438,442], &[598,602],\cr [602,606], &[606,609], &[609,610], &[615,618], &[638,642], &[642,645], &[645,646],\cr [651,654], &[663,665], &[741,742], &[759,762], &[805,806], &[822,826], &[826,830],\cr [854,861], &[902,903], &[935,938], &[969,970], &[986,987], &[1001,1002], &[1022,1023],\cr [1030,1034], &[1065,1066], &[1085,1086], &[1086,1090], &[1102,1105], &[1105,1106], &[1130,1131],\cr [1178,1182], &[1182,1185], &[1221,1222], &[1245,1246], &[1265,1266], &[1295,1298], &[1309,1310],\cr [1310,1311], &[1334,1335], &[1358,1362], &[1374,1378], &[1406,1407], &[1419,1426], &[1426,1434],\cr [1434,1435], &[1442,1443], &[1443,1446], &[1462,1463], &[1490,1491], &[1491,1495], &[1505,1506],\cr [1533,1534], &[1542,1545], &[1547,1551], &[1578,1581], &[1581,1582], &[1595,1598], &[1598,1599],\cr [1605,1606], &[1606,1614], &[1614,1615], &[1626,1630], &[1634,1635], &[1662,1670], &[1695,1698],\cr }$$ The sequence 102, 170, 230, 238, ... doesn't seem to be in the OEIS, although http://oeis.org/A215217 has the list of n such that n and n+1 are both products of three distinct primes.

Given any three distinct odd primes $a,b,c$, there are positive integers $s$ and $t$ such that $2as - bct = 1$, and then $2ax - bcy = 1$ for $x = s + bcn$, $y = t + 2an$ for any integer $n$. The arithmetic progressions $\{s+bcn: n \in {\mathbb N}\}$ and $\{t + 2an: n \in {\mathbb N}\}$ each have infinitely many primes by Dirichlet's theorem. I think it's likely that there are infinitely many $n$ for which both $x = s + bcn$ and $y = t + 2an$ are prime, so that $[bcy, 2ax]$ is in the list, but I suspect that proving this is far beyond our current capabilities.

Answer 2

Here's a list to get you started: $$[[230, 231], [255, 258], [285, 286], [429, 430], [434, 435], [609, 610], [645, 646], [741, 742], [805, 806], [902, 903], [969, 970], [986, 987], [1001, 1002], [1022, 1023], [1065, 1066], [1085, 1086], [1105, 1106], [1130, 1131], [1221, 1222], [1245, 1246], [1265, 1266], [1309, 1310], [1310, 1311], [1334, 1335], [1406, 1407], [1434, 1435], [1442, 1443], [1462, 1463], [1490, 1491], [1505, 1506], [1533, 1534], [1581, 1582], [1595, 1598], [1598, 1599], [1605, 1606], [1614, 1615], [1634, 1635], [1729, 1730], [1742, 1743], [1833, 1834], [1885, 1886], [1886, 1887], [1946, 1947], [2013, 2014], [2014, 2015], [2054, 2055], [2085, 2086], [2093, 2094], [2109, 2110], [2134, 2135]]$$

"Are there infinitely many?" Something around that...

Please note that I'm only posting this as an "answer" because it's too long for the comments - I don't have an actual proof that there are infinitely many but maybe this list will inspire you :)