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# Numbers fun: 11-11-11... another once-a-century palindrome!

From the numbers expert at the University of Portland comes another bit of fun with a palindrome date.

Aziz S. Inan, professor of Electrical Engineering shared this story with us:

November 11th is an annual United States holiday called Veterans Day to remember and honor military veterans. It’s also observed as Remembrance Day in Commonwealth countries to remember the members of their armed forces who died in line of duty since World War I.  This day commemorates the end of World War I, at the 11th hour of the 11th day of the 11th month of 1918!

November 11, this year (2011) to occur this coming Friday also has a special numerical property. If expressed as 11-11-11 (or simply, 111111), it is a very special once-a-century repunit (a number that contains only a repeated number, such as the digit 1) palindrome day! In recognition of this calendar day, I investigated its numerical properties as well as other such palindrome dates and here is what I found out:

Note that calendar day 111111 repeats once every century on November 11th in both date formats (month-day-year and day-month-year).

Now for the really wonky stuff (we're not kidding)

Also, Inan writes, note that if 111111 is expressed as 111111 = 37 x 7 x 3 x 11 x 13, the leftmost and the rightmost three numbers put side-by-side without the multiplication signs each constitute a palindrome number (3773 and 31113). In addition, 111111 equals the product of two palindrome numbers written as 111111 = 111 x 1001.

This year marks the 93rd anniversary of both Veterans and Remembrance Days where 93 = 31 x 3 where prime factors 31 and 3 put side-by-side without the multiplication sign forms 313, a palindrome number!111111 can be obtained from its largest prime factor 37 as follows: First, subtract 37 from its reverse which yields 36. (Note that 36 equal the square of the sum of the digits of 111111.) Then, split 36 into three consecutive numbers 11, 12 and 13 which add up to 36. Then, multiply numbers 11, 13, 37 and the reverse of 12! What comes out? Isn’t this fun? (By the way, prime factor 37 is also special because the leftmost and rightmost two digits of 1918 when World War I ended add up to 37! Also, the sum of the prime factors of 1918 (which are 2, 7 and 137) is 146 which equal twice the reverse of 37.)

In a non-leap year such as 2011, November 11th is the 315th day of the year where the digits of 315 given by 1, 3 and 5 are three consecutive odd numbers. In addition, 315 = 5 x 7 x 9 where 5, 7 and 9 are also three consecutive odd numbers. Also, reverse of 315 is 513 which equal 513 = 3 x 3 x 3 x 19 = 3 x 171 = 9 x 57 where its prime factors 3 and 19 add up to 22, a palindrome number. (Note that 22 = 11 + 11 which leads to 1111 without the in-between plus sign and 1111 corresponds to November 11!) Also, 9 + 57 = 66!

Assigning letters in the English alphabet numbers as A = 1, B = 2,…, Z = 26, the letters of “FRIDAY” and “ELEVEN” each add up to 63 which can be expressed as 3 x 7 x 3, a palindrome expression! Note that 5 x 63 = 315 (i.e., the 315th day)! Also, 63 is the reverse of 36, the square of the sum of the digits of 111111!

Also, the full date number of this Friday is 11-11-2011, or simply 11112011. If this number is split in the middle into numbers 1111 and 2011, these two numbers add up to 3122 = 2 x 7 x 223 where the sum of these three prime numbers equals 232, a palindrome number! Amazingly, the full date number of last week’s one-of-a-kind palindrome day November 2nd, 2011 expressed as 11022011 equals 72 x 113 x 132 where the three superscript numbers put side-by-side constitutes the very same palindrome number, 232!

Also, reverse of 11112011 is 11021111 which equal 1753 x 6287 where these two prime factors add up to 8040 = 2 x 2 x 2 x 3 x 5 x 67 where the sum of these prime factors equals 77! (Also, interestingly enough, by removing the prime factors 3 and 5 in the above product and replacing 2 x 2 x 2 x 67 by 2 x 2267 yields 4534, which corresponds to the difference of the prime factors 1753 and 6287!) Isn’t that something?

Other such palindrome days in this month are 11-1-11 (which is a repunit day), 11-2-11 (also a special one-of-a-kind palindrome day when written in full date format as 11-02-2011), 11-3-11, 11-4-11, 11-5-11, 11-6-11, 11-7-11, 11-8-11, 11-9-11, and 11-22-11. (Note that 11-1-11 can also be interpreted as 1-11-11 representing January 11, 2011. Also, a four-digit repunit day that occurred at the beginning of this year was January 1, 2011 represented by 1-1-11. In addition, January 10, 2011 written in full date format as 1-10-2011 was a palindrome day.) Among them, 112211 is interesting because it equals 101 x 11 x 101, a palindrome expression! Also, 11211 = 111 x 101, 11611 = 17 x 683 where 683 – 17 = 666, and 11811 = 3 x 31 x 127 where 3 + 31 + 127 = 161.

After this Friday, the next time 11-11-11 will occur 100 years from now in a palindrome (22nd) century in 2111. Interestingly enough, in 2111, 11-11-11 and eight-digit palindrome day 11-12-2111 will be two concurrent palindrome days!

Another interesting full palindrome day to occur 200 years from now 11 days after November 11 is 11-22-2211 or simply 11222211 which equals 101 times this Friday’s date, that is, 101 x 111111!

End of transmission.

And if this wasn’t enough intrigue involving numbers, check out Inan’s palindrome Website.

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