13th root

Extracting the 13th root of a number is a famous category for the mental calculation world records. The challenge consists of being given a large perfect 13th power (possibly over 100 digits) and asked to return the number that, when taken to the 13th power, equals the given number. For example, the 13th root of 8,192 is 2 and the 13th root of 96,889,010,407 is 7.

Properties of the challenge

Extracting the 13th root has certain properties. One is that the 13th root of a number is much smaller: a 13th root will have approximately 1/13th the number of digits. Thus, the 13th root of a 100-digit number only has 8 digits ^[1] and the 13th root of a 200-digit number will have 16 digits. Furthermore, the last digit of the 13th root is always the same as the last digit of the power. ^[1]

For the 13th root of a 100-digit number there are 7,992,563 possibilities, in the range 41,246,264 – 49,238,826. This is considered a relatively easy calculation. There are 393,544,396,177,593 possibilities, in the range 2,030,917,620,904,736 – 2,424,462,017,082,328, for the 13th root of a 200-digit number. This is considered a difficult calculation.

Records

100-digit numbers

The Guinness Book of World Records has published records for extracting the 13th root of a 100-digit number. ^[1] All world records for mentally extracting a 13th root have been for numbers with an integer root:

• The first record was 23 minutes by Herbert B. de Grote (Mexico) on October 5, 1970. ^[1]
• In 1981, Wim Klein (Netherlands) set a record of 88.8 seconds at the National Laboratory for High Energy Physics in Tsukuba, Japan. ^{[2] : 356}
• Gert Mittring calculated it in 39 seconds.
• Alexis Lemaire has broken this record with 13.55 seconds in 2002. ^[3] This is the last official world record for extracting the 13th root of a 100-digit number. ^[4]
• Mittring attempted to break this record with 11.8 seconds in November 2004. However, this was not counted as official as the Guinness Book of World Records had stopped recognizing records for root extraction of random numbers, due to the difficulty of standardizing the challenge. ^[3]
• Lemaire broke this record unofficially 6 times, twice within 4 seconds: the best was 3.625 seconds in December 2004. ^[5]

200-digit numbers

Lemaire has also set the first world record for the 13th root of a 200-digit number: 513.55 seconds and 742 attempts on April 6, 2005, and broken it with 267.77 seconds and 577 attempts on June 3, 2005. ^{[5] [6]}

• The same day, Lemaire has also set in front of official witnesses an unofficial record of 113 seconds and 40 attempts.
• On February 27, 2007, he set a world record of 1 minute and 47 seconds
• He broke this record on July 24, 2007, with a time of 1 minute and 17 seconds (77.99 seconds) at the Museum of History of Science, University of Oxford, UK. ^[7]
• Lemaire broke his record on November 15, 2007, with a time of 72.4 seconds
• Lemaire broke his record on December 10, 2007, with a time of 70.2 seconds

References

1. Smith, Steven Bradley (1983). The Great Mental Calculators: The Psychology, Methods, and Lives of Calculating Prodigies, Past and Present. New York: Columbia University Press. p. 129. ISBN   0231056419.
2. Hope, Jack A. (November 1985). "Unravelling the Mysteries of Expert Mental Calculation". Educational Studies in Mathematics. 16 (4): 355–374. doi:10.1007/BF00417192. JSTOR   3482445.
3. "All Hail The Nerd King". CBS News . 2004-11-24. Retrieved 2008-05-15.
4. Macura, Wiktor K.; Weisstein, Eric W. "13th Root". MathWorld . Retrieved 14 May 2025.
5. Bremner, Charles (2005-04-08). "What is the 13th root of . . . 836894668823695693983732866222 5645224726780466493836677 497357558157303507570408962 528802385783156837680293 493820105634336385559593151415 04151494907094190977044493 0566026840277186962415568 8082648640933?". The Times . London. Archived from the original on 2007-02-08. Retrieved 2008-05-15.
6. Bhattacharjee, Yudhijit (17 June 2005). "Milestones". Science . 308 (5729): 1739. doi:10.1126/science.308.5729.1739a.
7. "How does a human calculator do it?". BBC News Magazine . 30 July 2007. Retrieved 11 May 2025.