## Saturday, 30 March 2013

### SATURDAY, 30 MARCH 2013

Today is the $89^{th}$ day of the year.

$89$ is prime and the eleventh member of the Fibonacci sequence.

$89$ is a member of two primitive Pythagorean triples.
$(39, 80, 89)$
$(89, 3960, 3961)$

$5^2 + 8^2 = 25 + 64 = 89$
$2^2 + 2^2 + 9^2 = 4 + 4 + 81 = 89$
$2^2 + 6^2 + 7^2 = 4 + 36 + 49 = 89$
$3^2 + 4^2 + 8^2 = 9 + 16 + 64 = 89$

$(2 \times 89) + 1 = 179$ which is prime and, therefore, makes $89$ a Sophie Germain prime, see A005384.
$(2 \times 179) + 1 = 359$ which is prime
$(2 \times 359) + 1 = 719$ which is prime
$(2 \times 719) + 1 = 1,439$ which is prime
$(2 \times 1,439) + 1 = 2,879$ which is prime
$(2 \times 2,879) + 1 = 5,759 = 13 * 443$ which is not prime.
This gives us a Cunningham chain of $(89, 179, 359, 719, 1439, 2879)$ of length $6$. In fact $89$ is the smallest prime that starts a chain of this length, see A005602.

$2^{89} -1 = 618,970,019,642,690,137,449,562,111$ which is prime and, therefore, means that $89$ is a Mersenne Prime, see A000043.

T is the $89^{th}$ letter of the following, never-ending sentence:
"T is the first, fourth, eleventh, sixteenth, twenty-fourth, twenty-ninth, thirty-third, thirty-fifth, thirty-ninth,  ... letter in this sentence, not counting spaces or commas"
As beautifully illustrated in A005224, the sentence begins like this:

1234567890 1234567890 1234567890 1234567890 1234567890 Tisthefirs tfourthele venthsixte enthtwenty fourthtwen
tyninththi rtythirdth irtyfiftht hirtyninth fortyfifth
fortyseven thfiftyfir stfiftysix thfiftyeig hthsixtyse
condsixtyf ourthsixty ninthseven tythirdsev entyeighth
eightiethe ightyfourt heightynin thninetyfo urthninety
ninthonehu ndredfourt honehundre deleventho nehundreds
ixteenthon ehundredtw entysecond onehundred twentysixt
honehundre dthirtyfir stonehundr edthirtysi xthonehund