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ORIGINAL CHALLENGE:

13TH ROOT OF A 100-DIGIT NUMBER, ONE SINGLE TASK




The eleventh edition of the Guinness Book of World Records (1972:43) published that Herbert B. de Grote from Mexico successed, on October 5th 1970, in extracting mentally the 13th root from a 100-digit number in 23 minutes.

This record was taken by the "Human Computer" of CERN, Wim Klein from Netherlands. Wim Klein worked in fact at the CERN as a human calculator checking the results of the computers, and it happened sometimes that he beat the machine(not as powerful as the ones of today)
His time of calculation was far faster than De Grote's. He improved it many times: the first official one was more than 5 minutes and the last one was 1mn28s. He also tried other roots like the 73rd root from a 507-digit number, the 7th root from a 63-digit number and the 19th root from a 133-digit number
However, the 13th root of 100-digit number stands as the most published world record in the area of roots mental extraction. The 73rd root from a 500 or 507-digit number is an easier task, mainly due to the fact (extremely important) that the number of possibilities is far lower:
A 13th root is generally harder than a cube root and what is the most important in such cases is the number of possibilities:
the 13th roots of 100-digit number belong to 41246264-49238826: 7992563 possibilities. We will see in the Apocalypse section amazing things about this.
the 73rd roots of 500/507 digits have 219471/273696 possibilities.




The most famous time is 1mn28s broken on Tuesday, April 7th 1981 in the laboratory of high energies physics at Tsukuba, Japan. He correctly extracted the 13th root from this 100-digit number:


88008443440489299575219015772236417859411720052615
65487280650870412023307854274990144578442271602817

His answer was 48757377

According to the Saxonia record club, the time was firstly broken by the famous lightening calculator Gert Mittring who would have extracted it in 39.0s on May 26th, 1988

Guinness mistakes are not uncommon and they published once the time 0.15s by Jaime Garcia Serrano(Colombia), misunderstanding how to measure mental calculation records(the time must stop at the end of the answer not at the beginning!) Probably this is also the reason why they published a record like multiplying two 13-digit numbers in only 28s (on June 18th, 1980) mentionning the obvious objections of the specialists. I have a statement by Guinness which asserts that on December 5th 2000 the Guinness world record was still 1mn 28s.

But the 1mn28.8s/39s were broken by the French Alexis Lemaire with the time 13.55s.


He gave the correct answer of the following problem:

29288115834875201060553567352783652122196502020937
13928425510086152669633464222587770308279739304053

The correct answer is 44800613

HISTORY OF THIS RECORD

BEGINNER CHALLENGE:FASTEST OPPORTUNITY

Name Nationality Time in secondes Place, Country Date
Herbert B. de Grote Mexican nearly 1380 Mexico October 5th, 1970
Wilhelm Klein Dutch 322 Amsterdam, Netherlands September 19th 1975
Wilhelm Klein Dutch 231 Stockholm, Sweden November 8th, 1978
Wilhelm Klein Dutch 205 Providence, Rhodes Island September 1979
Wilhelm Klein Dutch 186 Paris, France November 1979
Wilhelm Klein Dutch 165 Leiden March 1980
Wilhelm Klein Dutch 129 London, England May 6th, 1980
Wilhelm Klein Dutch 128 Berlin, Germany November 10th, 1980
Wilhelm Klein Dutch 116 November 13th, 1980
Wilhelm Klein Dutch 88.8 Tsukuba, Japan April 7th, 1981
(Gert Mittring?) (German?) (39.0?) (Germany?) (May 26th, 1988?*)
(Alexis Lemaire?) (French?) (24.48?**) (Villers-Marmery, France?) (May 10th, 2002?)
Alexis Lemaire French 13.55 Villers-Marmery, France May 10th, 2002


*Date given by Gert Mittring , NOT ENOUGH DOCUMENTATION
**Time measured by the witnesses/the video without attestation NOT ENOUGH DOCUMENTATION

***

Because of their own lack of accuracy around the rules, the Guinness World Records book is no longer able to verify and to accept a new record in the category 13th root

the real official task must be realised with 9 exact calculations: the first one ends in 1, the second one in 2, ..., the last one in 9

See the section "world and universe records" for further explanations


BEGINNER CHALLENGE:FASTEST CALCULATOR


NO ENTRY YET

Methods


SOURCE:"the Great Mental Calculators (...)" ,CHAPTER 13, Steven Smith, 1983

Thanks to Oleg Stepanov for his publication on the web


Klein's methods for extracting 13th roots can be illustrated with the following number:
14762420839370760705665953772022217870318956930659 27236796230563061507768203333609354957218480390144

The first five digits of the root are fixed through the use of logarithms, Klein has memorized to five places the logs of the integers up to 150; this, coupled with his ability to factor large numbers, allows him to approximate the log of the first five digits of the power, which is usually sufficient to determine the first five digits of the root, though, as he says "the fifth digit is a bit chancy." ' Klein began by factoring 1,476 into 36 times 41 and taking the (decimal) log of each: log 36 = 1.55630 and log 41 = 1.61278; adding the mantissas yields 0.16908, but this is, of course, too little. Through various interpolations Klein estimated the mantissa of the log of 147,624 as 0.16925 (it is more nearly 0.16916). Klein now had an approximation of the log of the 100-digit number above - 99.16925. This must be divided by 13 to obtain the log of the 13th root. Since 99=13X7 with a remainder of 8, to obtain the mantissa of the antilog of the 13th root he divided 8.16925 by 13, which is approximately 0.62840. He estimated the antilog to be about halfway between 4.2 and 4.3 and decided to try 4.25. The result was exact, so the first five digits of the root should be 42500, as indeed they are. It is now necessary to determine the last three digits of the root. This he does from an examination of the last three digits of the power. In the case of odd powers, these uniquely determine the last three digits of the root, but in the case of even roots, like this one, this method yields four possibilities; in the case of 144 they are 014, 264, 514, and 764. (The choices always differ by 250.) To select the correct one Klein divides the original number by 13 and retains the remainder. In the case of 13th roots, the root remainder and the power remainder must be the same. The power remainder is 7; only 764 as the final three digits of the root will yield 7 as the remainder. Thus the 13th root is determined to be 42,500,764. As an example of an odd root take:

75185285487713563581947553291145079861723813162341 53935861550997297991815299022662358976308065985831

The first five digits of the power are 75185, which is nearly 7519, and 7519 is 73 times 103. The mantissa of the log of 73 is 0.86332 and that of 103 is 0.01284. Their sum is 0.87616. Dividing 8.87616 by 13 yields 0.68278. This falls between the mantissas of the logs of 48 and 49, but is much closer to 48. Since 481 is 13 (mantissa 0.11394) times 37 (mantissa 0.56820), the mantissa of its log will be 0.68214; dose, but still a bit low; 4,816 can be factored into 16 (mantissa 0.20412) times 7 (mantissa 0.84510) times 43 (mantissa 0.63347). This gives a mantissa of 0.68269. Then 4,818 factors into 66 (mantissa 0.81954) times 73 (mantissa 0.86332), which yields a mantissa of 0.68286. Thus, in the interpolation we want 9/17 of 20 which is about 10 1/2. The first five digits of the root should be 48170 (48160 + 10). This, in fact, is correct. When Klein actually did the calculation he made a minor error (he was looking for the antilog of 0.68277 instead of 0.68278) and first took 48169 for the first five digits of the root. In this case, however, since the root is odd, the last three digits are uniquely fixed-since the power ends in 831, the root must end in 311. Upon dividing the power by 13 Klein got a remainder of 7. But dividing 48,169,311 by 13 gives a remainder of 8. To make these two remainders come into line he changed his solution to 48,170,311, which is correct.


Gert Mittring's method for the last 3 digits


This method was suggested by Gert Mittring when he calculated the 137th root from a 1000-digit number; we can illustrate it with the 13th root from a 100-digit number

00 01 02 03 04
1 001 931 461 591 321
3 323 253 383 713 243
7 407 937 267 397 327
9 329 059 189 719 649


Example of meaning: 4713=...327
We can write pw=327, rt=047

Mittring' rules

RULE A: if pw2=pw1+650 then rt2=rt1+050
RULE B: if pw2=pw1+300 then rt2=rt1+100

Example if pw2=651, pw1=pw2-650=001, rt1=001, rt2=001+050=051 : 51^13=...651

RULE C: if pw2=pw1+130 then rt2=rt1+010
RULE D: if pw1+pw2=650 then rt1+rt2=050

Alexis Lemaire equivalence relation with 100 digits



In the case of first category of endings(roots endings 1,3,7 and 9) Alexis Lemaire was the first to claim to be the discoverer of the following relation:

X13=Y <=> X=Y77


This is the Alexis Lemaire equivalence relation applied to the last 4 digits of the 13th root = 77th power



Discussion group about the 13th root


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