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Part of 2019 Irish Math Olympiad

Problems(1)

2019 is smallest no, sum of fourth powers of 5 int, sum of 6 consecutive

Source: 2019 Irish Mathematical Olympiad paper 2 p6

10/5/2020
The number 20192019 has the following nice properties: (a) It is the sum of the fourth powers of fuve distinct positive integers. (b) It is the sum of six consecutive positive integers. In fact, 2019=14+24+34+54+642019 = 1^4 + 2^4 + 3^4 + 5^4 + 6^4 (1) 2019=334+335+336+337+338+3392019 = 334 + 335 + 336 + 337 + 338 + 339 (2) Prove that 20192019 is the smallest number that satis es both (a) and (b). (You may assume that (1) and (2) are correct!)
number theorySum of powersconsecutive