109 times 19

109 and 19 are both prime numbers. The product of two primes form a class of numbers that are useful in cryptography. The product of 109 and 19 is 2071.

I enjoy fun numbers like that. Most people, especially computer nerds, like powers of two — for example 1024 or 2048. Computer displays often come in sizes that are a power of two. Or are midway between two powers of two. For example 768, a common size vertical screen resolution, is 256 (2^8) more than 512 (2^9).

Powers of two are rather boring any more. Multiples of two primes are fun… I wonder if they’re pretty common or not.

Here is a list of the first 10000 primes should you ever need to know that.

To answer my question about how many multiples of two primes there are…. Well, here is a lua program that calculates all the values less than 3000 that are the multiple of two primes.


primes = {
2, 3, 5, 7, 11, 13, 17, 19, 23, 29,
31, 37, 41, 43, 47, 53, 59, 61, 67, 71,
73, 79, 83, 89, 97, 101, 103, 107, 109, 113,
127, 131, 137, 139, 149, 151, 157, 163, 167, 173,
179, 181, 191, 193, 197, 199, 211, 223, 227, 229,
233, 239, 241, 251, 257, 263, 269, 271, 277, 281,
283, 293, 307, 311, 313, 317, 331, 337, 347, 349,
353, 359, 367, 373, 379, 383, 389, 397, 401, 409,
419, 421, 431, 433, 439, 443, 449, 457, 461, 463,
467, 479, 487, 491, 499, 503, 509, 521, 523, 541,
547, 557, 563, 569, 571, 577, 587, 593, 599, 601,
607, 613, 617, 619, 631, 641, 643, 647, 653, 659,
661, 673, 677, 683, 691, 701, 709, 719, 727, 733,
739, 743, 751, 757, 761, 769, 773, 787, 797, 809,
811, 821, 823, 827, 829, 839, 853, 857, 859, 863,
877, 881, 883, 887, 907, 911, 919, 929, 937, 941,
947, 953, 967, 971, 977, 983, 991, 997, 1009, 1013,
1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069,
1087, 1091, 1093, 1097, 1103, 1109, 1117, 1123, 1129, 1151,
1153, 1163, 1171, 1181, 1187, 1193, 1201, 1213, 1217, 1223,
1229, 1231, 1237, 1249, 1259, 1277, 1279, 1283, 1289, 1291,
1297, 1301, 1303, 1307, 1319, 1321, 1327, 1361, 1367, 1373,
1381, 1399, 1409, 1423, 1427, 1429, 1433, 1439, 1447, 1451,
1453, 1459, 1471, 1481, 1483, 1487, 1489, 1493, 1499, }

thePairs = {}
ti = table.insert

for _, a in ipairs( primes ) do
for _, b in ipairs( primes ) do
thePairs[ a*b ] = true
end
end

thePairs2 = {}
for k, _ in pairs( thePairs ) do
ti( thePairs2, k )
end
table.sort( thePairs2 )
for _, x in ipairs( thePairs2 ) do
print( x )
if x > 3000 then
break
end
end

They’re actually pretty common. For example, here are all the years I’ve lived through that were the multiple of two primes:

1977
1981
1982
1983
1985
1991
1994
2005

(I’ll spare you the full listing: There are 842 multiples of two primes less than 3000. No make that 841, I had an off-by-one error in my original calculation.)

The next one isn’t until 2018. A gap of 13 years seems like a big one… But the biggest gap is in 2681, when it will be twenty years before another year which is a multiple of two primes comes along.

My grandfather used to search for decades of numbers where the values for that decade ending in 1,3,7 and 9 were all prime. For example, 11,13, 17 and 19 is the first such example. Mom bought him a programable HP calculator in the early 80s to help his search. So, interest in numbers is nothing new in my family.