My Favorite Prime Number with Four Divisors
A prank I recommend to readers is to use the number 91 when a group situation calls for a random prime number. If done subtly enough, a decent portion of mathematicians will believe you. Granted, its not a particularly funny prank…
So why 91? Look at the main ways to quickly check primality of two digit numbers:
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Is it even?
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Does it end in 5?
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Do its digits add up to a multiple of 3?
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Are the two digits the same?
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Is it a square?
For these, 91 is the only composite number that all of them miss (being 7 times 13). Most mathematicians run those checks in their head, or (more likely) they are pretty good at spotting immediately numbers which fail the above checks. Anything that passes these checks at least ’smells’ prime.
This fact is also useful for something other than lame pranks. It means that you can know for sure which two digit numbers are prime, simply by adding one last check to the list:
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Is it 91?
Its not super useful, but it does cut down on the amount of time it takes to check numbers like 83. It smells prime, and its not 91, so its prime! Its like the quickest Sieve method imaginable.
February 11, 2008 at 11:51 am
Check for 7s, A little known by fairly useful divisibility test: Double the 1s digit and subtract the result from the previous digits: 9- 2(1)=7. If the result is divisible by 7, then the number is divisible by 7.
Two more examples, to check 343 (yeah I know it is 7×7x7), 34-6-28. Meanwhile, 2- 2(8)=-14.
2401 = (7×7x7×7):
240-2(1)=238 = 7×34.
23 -2(8)=23-16=7.
For big (between 100-9800) that end in 1s 3s or 7s,
when other divisibility tests immediately fail, check for numbers ending in 3, 7, or 9 that are near the square roots. Notice this works for 91 as well.
February 11, 2008 at 11:53 am
GRRH typo: 34-6=28.
February 11, 2008 at 3:18 pm
@ carter
do you have a proof for that? if so, its a fairly interesting divisibility property, and I wonder if it could be generalized.
February 11, 2008 at 4:58 pm
Great post! I, too, have enjoyed the fact that 91 “smells” prime, but isn’t. However, I’ve never considered using that fact for the underhanded motives of fooling others. Now I will.
A couple comments mentioned divisibility tests. I’d like to point out a paper of mine, “Stupid Divisibility Tricks,” available at my website. I hope you enjoy!
February 11, 2008 at 5:39 pm
Joe-
Think of it instead as subtracting 21 times the units digit from the original number. The resulting number is a multiple of 10, so you can divide it by ten to get the final number. Neither of the two operations changed divisibilty by 7.
This is also how one can think of the divisibility by 3 and 11 tests (checking the sum of the digits, and checking the alternating sum of the digits), since these amount to a series of operations which don’t change divisibility. This all can be said a bit slicker in modular arithmetic, but there’s no need.
February 15, 2008 at 12:17 am
You know about 57, the so-called Grothendieck prime? (You can tell by adding the digits that it is a multiple of 3). A quick google on the phrase found several descriptions of the incident, as described here: http://www.ams.org/notices/200410/fea-grothendieck-part2.pdf