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Question Number 32610 by naka3546 last updated on 29/Mar/18

2.  Find  the  number  of  ordered  triples  (a, b, c)  of  integers  satisfying    0 ≤  a, b, c  ≤  1000   for  which                            a^3  + b^3  + c^3   ≡  3abc + 1   (mod  1001)

$$\mathrm{2}.\:\:{Find}\:\:{the}\:\:{number}\:\:{of}\:\:{ordered}\:\:{triples}\:\:\left({a},\:{b},\:{c}\right)\:\:{of}\:\:{integers}\:\:{satisfying}\:\:\:\:\mathrm{0}\:\leqslant\:\:{a},\:{b},\:{c}\:\:\leqslant\:\:\mathrm{1000}\:\:\:{for}\:\:{which} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{a}^{\mathrm{3}} \:+\:{b}^{\mathrm{3}} \:+\:{c}^{\mathrm{3}} \:\:\equiv\:\:\mathrm{3}{abc}\:+\:\mathrm{1}\:\:\:\left({mod}\:\:\mathrm{1001}\right)\: \\ $$

Commented by Rasheed.Sindhi last updated on 30/Mar/18

(1,0,0),(0,1,0) & (0,0,1) are satisfying     (Not complete.)

$$\left(\mathrm{1},\mathrm{0},\mathrm{0}\right),\left(\mathrm{0},\mathrm{1},\mathrm{0}\right)\:\&\:\left(\mathrm{0},\mathrm{0},\mathrm{1}\right)\:\mathrm{are}\:\mathrm{satisfying} \\ $$$$\:\:\:\left(\mathrm{Not}\:\mathrm{complete}.\right)\:\: \\ $$$$ \\ $$

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