Thursday, December 5, 2013

Weird Math...

What do the two numbers, 70 and

26,963,672,211,957,831,828,322,834,071,143,299,817,754, 720,290,127,404,079,937,026,385,368,922,075,196,690,720,690,562,498,337,038,657,263, 353,255,952,256,005,850,803,053,091,152,216,128,172,198,270,512,414,580,092,743,322, 379,544,478,286,025,897,899,890,351,444,085,611,625,835,160,270,418,964,124,507,243, 890,975,821,522,176,465,361,680,177,670,297,930,314,037,850,339,675,559,057,554,452, 347,547,946,165,134,639,879,111,112,583,151,946,671,967,876,920,506,598,818,088,728, 910,330,021,016,856,674,391,763,268,224,262,067,132,913,691,721,407,174,127,885,521, 288,146,239,271,038,154,486,086,650,600,357,888  ...have in common?

They are both "weird" numbers… and I mean that, in a technical way! 70 is the smallest "weird" number and that second monstrosity is, to date, the largest known (of an infinite number) of weird numbers, at 226 digits. It was found by these Central Washington University folks:
also see here:

"Weird" numbers are those natural numbers whose divisors add up to more than the number itself, and for which NO selection of divisors sum exactly to the original number [for example, for 70, the divisors are 1, 2, 5, 7, 10, 14, and 35, which sum to 74, and no possible combination adds exactly to 70]. The student group originally discovered the first new weird number in over three decades, with a 72-digit find, before eventually reaching the above record.  Per the article, "a better understanding of weird numbers leads to a better understanding of factorization, which is the basis of all modern cryptography." [in case you were wondering of what possible use this could be!]

Here is a sequence of weird numbers from the OEIS directory:

70, 836, 4030, 5830, 7192, 7912, 9272, 10430, 10570, 10792, 10990, 11410, 11690, 12110, 12530, 12670, 13370, 13510, 13790, 13930, 14770, 15610, 15890, 16030, 16310, 16730, 16870, 17272, 17570, 17990, 18410, 18830, 18970, 19390, 19670

Interesting that all of these, with the single exception of 836, end with a "2" or a zero, yet the new record find ends with an 8. -- I have no idea what the distribution of end-digits is for the full panoply of currently-known weird numbers??? (It is also not known with certainty if ANY odd weird numbers exist... but if they do, they must be very, VERY large!)

[I don't know if it's even possible to explain at a layperson level, but if someone in-the-know wants to try and explain in the comments what sort of method/algorithm one employs to discover weird numbers of such length (or alternatively how one verifies such a number) I'd be curious to hear it.]

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