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.

2 thoughts on “109 times 19

  1. Welcome back! Your grandfather would be impressed that you remembered his research! Unfortunately, he never succeeded in learning to program the calculator that I gave him. He used an older HP which had about given up the ghost by 1981.

  2. Welcome back! Your grandfather would be impressed that you remembered his research! Unfortunately, he never succeeded in learning to program the calculator that I gave him. He used an older HP which had about given up the ghost by 1981.

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