## Friday, 8 March 2013

### FRIDAY, 8 MARCH 2013

Today is the $67^{th}$ day of the year.

$67$ is a prime number and is the nineteenth in the sequence of prime numbers, see A000040.

$67$ is a toothpick number, see A139250 and a nice illustration here.

Consider the powers of $2$ modulo 67.
$2^0 = 1 = 1$
$2^{1} = 2 * 2^{0} = 2 * 1 = 2$
$2^{2} = 2 * 2^{1} = 2 * 2 = 4$
$2^{3} = 2 * 2^{2} = 2 * 4 = 8$
$2^{4} = 2 * 2^{3} = 2 * 8 = 16$
$2^{5} = 2 * 2^{4} = 2 * 16 = 32$
$2^{6} = 2 * 2^{5} = 2 * 32 = 64$
$2^{7} = 2 * 2^{6} = 2 * 64 = 61$
$2^{8} = 2 * 2^{7} = 2 * 61 = 55$
$2^{9} = 2 * 2^{8} = 2 * 55 = 43$
$2^{10} = 2 * 2^{9} = 2 * 43 = 19$
$2^{11} = 2 * 2^{10} = 2 * 19 = 38$
$2^{12} = 2 * 2^{11} = 2 * 38 = 9$
$2^{13} = 2 * 2^{12} = 2 * 9 = 18$
$2^{14} = 2 * 2^{13} = 2 * 18 = 36$
$2^{15} = 2 * 2^{14} = 2 * 36 = 5$
$2^{16} = 2 * 2^{15} = 2 * 5 = 10$
$2^{17} = 2 * 2^{16} = 2 * 10 = 20$
$2^{18} = 2 * 2^{17} = 2 * 20 = 40$
$2^{19} = 2 * 2^{18} = 2 * 40 = 13$
$2^{20} = 2 * 2^{19} = 2 * 13 = 26$
$2^{21} = 2 * 2^{20} = 2 * 26 = 52$
$2^{22} = 2 * 2^{21} = 2 * 52 = 37$
$2^{23} = 2 * 2^{22} = 2 * 37 = 7$
$2^{24} = 2 * 2^{23} = 2 * 7 = 14$
$2^{25} = 2 * 2^{24} = 2 * 14 = 28$
$2^{26} = 2 * 2^{25} = 2 * 28 = 56$
$2^{27} = 2 * 2^{26} = 2 * 56 = 45$
$2^{28} = 2 * 2^{27} = 2 * 45 = 23$
$2^{29} = 2 * 2^{28} = 2 * 23 = 46$
$2^{30} = 2 * 2^{29} = 2 * 46 = 25$
$2^{31} = 2 * 2^{30} = 2 * 25 = 50$
$2^{32} = 2 * 2^{31} = 2 * 50 = 33$
$2^{33} = 2 * 2^{32} = 2 * 33 = 66$
$2^{34} = 2 * 2^{33} = 2 * 66 = 65$
$2^{35} = 2 * 2^{34} = 2 * 65 = 63$
$2^{36} = 2 * 2^{35} = 2 * 63 = 59$
$2^{37} = 2 * 2^{36} = 2 * 59 = 51$
$2^{38} = 2 * 2^{37} = 2 * 51 = 35$
$2^{39} = 2 * 2^{38} = 2 * 35 = 3$
$2^{40} = 2 * 2^{39} = 2 * 3 = 6$
$2^{41} = 2 * 2^{40} = 2 * 6 = 12$
$2^{42} = 2 * 2^{41} = 2 * 12 = 24$
$2^{43} = 2 * 2^{42} = 2 * 24 = 48$
$2^{44} = 2 * 2^{43} = 2 * 48 = 29$
$2^{45} = 2 * 2^{44} = 2 * 29 = 58$
$2^{46} = 2 * 2^{45} = 2 * 58 = 49$
$2^{47} = 2 * 2^{46} = 2 * 49 = 31$
$2^{48} = 2 * 2^{47} = 2 * 31 = 62$
$2^{49} = 2 * 2^{48} = 2 * 62 = 57$
$2^{50} = 2 * 2^{49} = 2 * 57 = 47$
$2^{51} = 2 * 2^{50} = 2 * 47 = 27$
$2^{52} = 2 * 2^{51} = 2 * 27 = 54$
$2^{53} = 2 * 2^{52} = 2 * 54 = 41$
$2^{54} = 2 * 2^{53} = 2 * 41 = 15$
$2^{55} = 2 * 2^{54} = 2 * 15 = 30$
$2^{56} = 2 * 2^{55} = 2 * 30 = 60$
$2^{57} = 2 * 2^{56} = 2 * 60 = 53$
$2^{58} = 2 * 2^{57} = 2 * 53 = 39$
$2^{59} = 2 * 2^{58} = 2 * 39 = 11$
$2^{60} = 2 * 2^{59} = 2 * 11 = 22$
$2^{61} = 2 * 2^{60} = 2 * 22 = 44$
$2^{62} = 2 * 2^{61} = 2 * 44 = 21$
$2^{63} = 2 * 2^{62} = 2 * 21 = 42$
$2^{64} = 2 * 2^{63} = 2 * 42 = 17$
$2^{65} = 2 * 2^{64} = 2 * 17 = 34$

Every number between $1$ and $66$ appears just once. There are no repetitions. This makes $2$ a primitive root of, the prime, $67$. See A001122.

This is the above list in modulo $67$ order.

$2^{0} = 1$
$2^{1} = 2 * 2^{0} = 2 * 1 = 2$
$2^{39} = 2 * 2^{38} = 2 * 35 = 3$
$2^{2} = 2 * 2^{1} = 2 * 2 = 4$
$2^{15} = 2 * 2^{14} = 2 * 36 = 5$
$2^{40} = 2 * 2^{39} = 2 * 3 = 6$
$2^{23} = 2 * 2^{22} = 2 * 37 = 7$
$2^{3} = 2 * 2^{2} = 2 * 4 = 8$
$2^{12} = 2 * 2^{11} = 2 * 38 = 9$
$2^{16} = 2 * 2^{15} = 2 * 5 = 10$
$2^{59} = 2 * 2^{58} = 2 * 39 = 11$
$2^{41} = 2 * 2^{40} = 2 * 6 = 12$
$2^{19} = 2 * 2^{18} = 2 * 40 = 13$
$2^{24} = 2 * 2^{23} = 2 * 7 = 14$
$2^{54} = 2 * 2^{53} = 2 * 41 = 15$
$2^{4} = 2 * 2^{3} = 2 * 8 = 16$
$2^{64} = 2 * 2^{63} = 2 * 42 = 17$
$2^{13} = 2 * 2^{12} = 2 * 9 = 18$
$2^{10} = 2 * 2^{9} = 2 * 43 = 19$
$2^{17} = 2 * 2^{16} = 2 * 10 = 20$
$2^{62} = 2 * 2^{61} = 2 * 44 = 21$
$2^{60} = 2 * 2^{59} = 2 * 11 = 22$
$2^{28} = 2 * 2^{27} = 2 * 45 = 23$
$2^{42} = 2 * 2^{41} = 2 * 12 = 24$
$2^{30} = 2 * 2^{29} = 2 * 46 = 25$
$2^{20} = 2 * 2^{19} = 2 * 13 = 26$
$2^{51} = 2 * 2^{50} = 2 * 47 = 27$
$2^{25} = 2 * 2^{24} = 2 * 14 = 28$
$2^{44} = 2 * 2^{43} = 2 * 48 = 29$
$2^{55} = 2 * 2^{54} = 2 * 15 = 30$
$2^{47} = 2 * 2^{46} = 2 * 49 = 31$
$2^{5} = 2 * 2^{4} = 2 * 16 = 32$
$2^{32} = 2 * 2^{31} = 2 * 50 = 33$
$2^{65} = 2 * 2^{64} = 2 * 17 = 34$
$2^{38} = 2 * 2^{37} = 2 * 51 = 35$
$2^{14} = 2 * 2^{13} = 2 * 18 = 36$
$2^{22} = 2 * 2^{21} = 2 * 52 = 37$
$2^{11} = 2 * 2^{10} = 2 * 19 = 38$
$2^{58} = 2 * 2^{57} = 2 * 53 = 39$
$2^{18} = 2 * 2^{17} = 2 * 20 = 40$
$2^{53} = 2 * 2^{52} = 2 * 54 = 41$
$2^{63} = 2 * 2^{62} = 2 * 21 = 42$
$2^{9} = 2 * 2^{8} = 2 * 55 = 43$
$2^{61} = 2 * 2^{60} = 2 * 22 = 44$
$2^{27} = 2 * 2^{26} = 2 * 56 = 45$
$2^{29} = 2 * 2^{28} = 2 * 23 = 46$
$2^{50} = 2 * 2^{49} = 2 * 57 = 47$
$2^{43} = 2 * 2^{42} = 2 * 24 = 48$
$2^{46} = 2 * 2^{45} = 2 * 58 = 49$
$2^{31} = 2 * 2^{30} = 2 * 25 = 50$
$2^{37} = 2 * 2^{36} = 2 * 59 = 51$
$2^{21} = 2 * 2^{20} = 2 * 26 = 52$
$2^{57} = 2 * 2^{56} = 2 * 60 = 53$
$2^{52} = 2 * 2^{51} = 2 * 27 = 54$
$2^{8} = 2 * 2^{7} = 2 * 61 = 55$
$2^{26} = 2 * 2^{25} = 2 * 28 = 56$
$2^{49} = 2 * 2^{48} = 2 * 62 = 57$
$2^{45} = 2 * 2^{44} = 2 * 29 = 58$
$2^{36} = 2 * 2^{35} = 2 * 63 = 59$
$2^{56} = 2 * 2^{55} = 2 * 30 = 60$
$2^{7} = 2 * 2^{6} = 2 * 64 = 61$
$2^{48} = 2 * 2^{47} = 2 * 31 = 62$
$2^{35} = 2 * 2^{34} = 2 * 65 = 63$
$2^{6} = 2 * 2^{5} = 2 * 32 = 64$
$2^{34} = 2 * 2^{33} = 2 * 66 = 65$
$2^{33} = 2 * 2^{32} = 2 * 33 = 66$