Sum of three cubes for 42 finally solved
09-15-2019, 04:20 PM (This post was last modified: 09-15-2019 04:39 PM by Albert Chan.)
Post: #7
 Albert Chan Senior Member Posts: 1,676 Joined: Jul 2018
RE: Sum of three cubes for 42 finally solved
(09-14-2019 09:14 PM)Gerson W. Barbosa Wrote:  34 significant digits are not enough for the task. That would require at least 50.

We can split the numbers, like this:

a = −80538738812075974 = −80538739e9 + 187924026 = a1 + a0
b = +80435758145817515 = +80435758e9 + 145817515 = b1 + b0
c = +12602123297335631 = +12602123e9 + 297335631 = c1 + c0

a³ + b³ + c³
= (a1 + a0)³ + (b1 + b0)³ + (c1 + c0
= (a1³ + b1³ + c1³) + 3*(a0a1² + b0b1² + c0c1²) + 3*(a1a0² + b1b0² + c1c0²) + (a0³ + b0³ + c0³)

= -6628842934503040000000000000000000000000000
﻿ ﻿ ﻿+6628842934562563667007015000000000000000000
﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿-59523703031106040674633000000000
﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿ ﻿+36024091040674633000000042
= 42

For calculators with precision of 12 decimal digits, we can do mod test:
PHP Code:
a = {-80538738812, -075974}   -- a^3 ~ -5.2241E+50b = { 80435758145,  817515}   -- b^3 ~ +5.2041E+50c = { 12602123297,  335631}   -- c^3 ~ +0.0200E+50function cube(x, n)    x = (x[1]%n * 1e6 + x[2]) % n    return x * x % n * x % nendmodtest = function(n) return (cube(a,n) + cube(b,n) + cube(c,n)) % n end

lua> modn = {999907,999917,999931,999953,999959,999961,999979,999983,1e6}
lua> for _, n in ipairs(modn) do print(modtest(n), '(mod ' .. n .. ')') end

42 (mod 999907)
42 (mod 999917)
42 (mod 999931)
42 (mod 999953)
42 (mod 999959)
42 (mod 999961)
42 (mod 999979)
42 (mod 999983)
42 (mod 1e+006)

Except for 1e6, all mods are primes, we get a³ + b³ + c³ ≡ 42 (mod ~ 1e54)
Since 1e54 ≫ |sum of three cubes| , we get a³ + b³ + c³ = 42
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 Messages In This Thread Sum of three cubes for 42 finally solved - Gerson W. Barbosa - 09-14-2019, 03:20 PM RE: Sum of three cubes for 42 finally solved - Tugdual - 09-14-2019, 05:03 PM RE: Sum of three cubes for 42 finally solved - DA74254 - 09-14-2019, 06:11 PM RE: Sum of three cubes for 42 finally solved - Gerson W. Barbosa - 09-14-2019, 09:14 PM RE: Sum of three cubes for 42 finally solved - Albert Chan - 09-15-2019 04:20 PM RE: Sum of three cubes for 42 finally solved - Jaco@cocoon-creations.com - 09-15-2019, 12:37 PM RE: Sum of three cubes for 42 finally solved - DA74254 - 09-15-2019, 07:19 PM RE: Sum of three cubes for 42 finally solved - Jaco@cocoon-creations.com - 09-15-2019, 11:51 PM RE: Sum of three cubes for 42 finally solved - Paul Dale - 09-15-2019, 06:29 AM RE: Sum of three cubes for 42 finally solved - StephenG1CMZ - 09-16-2019, 07:28 AM RE: Sum of three cubes for 42 finally solved - DA74254 - 09-16-2019, 05:38 PM RE: Sum of three cubes for 42 finally solved - Tugdual - 09-27-2019, 10:52 PM

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