The smallest squares containingk 98's :
9801 = 99²,   2985984 = 1728²,   9846989824 = 99232²,
19819849898601 = 4451949²,   9870269898989824 = 99349232².

98 is the sum of m squares (m = 2,3,4,...,84).

98² = 9604, a square with different digits.

98² = 9604, 96 + 0 + 4 = 10².

98² is the 6th square which is the sum of 2 cubes : 7³ + 21³,
98² is the 6th square which is the sum of 3 cubes : 1³ + 14³ + 19³.

98² = 7³ + 21³ = 1³ + 14³ + 19³ = 7⁴ + 7⁴ + 7⁴ + 7⁴.

98² = 24² + 48² + 82² = 28² + 84² + 42².

98² = (2² + 3)(37² + 3).

5² + 29² + 53² + 77² = 98².

98^k + 212^k + 305^k + 346^k are squares for k = 1,2,3 (31², 517², 8959²).
49^k + 98^k + 170^k + 212^k are squares for k = 1,2,3 (23², 293², 3937²).
34^k + 82^k + 98^k + 110^k are squares for k = 1,2,3 (18², 172², 1692²).
10^k + 74^k + 98^k + 142^k are squares for k = 1,2,3 (18², 188², 2052²).
490^k + 1022^k + 1274^k + 6818^k are squares for k = 1,2,3 (98², 7028², 565852²).
665^k + 1421^k + 2401^k + 5117^k are squares for k = 1,2,3 (98², 5866², 388570²).
994^k + 1470^k + 2338^k + 4802^k are squares for k = 1,2,3 (98², 5628², 357308²).

Komachi equations:
98² = 12² * 3² + 4² * 5² + 6² - 7² + 89² = 98² + 7² - 6² - 5² + 4² - 3² + 2² + 1²
  = 98² - 7² + 6² + 5² - 4² + 3² - 2² - 1² = 98² + 7² - 6² / 54² * 3² * 21²
  = 98² + 7² / 6² * 54² - 3² * 21² = 98² + 7² / 6² * 54² / 3² - 21²
  = 98² - 7² + 6² / 54² * 3² * 21² = 98² - 7² / 6² * 54² + 3² * 21²
  = 98² - 7² / 6² * 54² / 3² + 21².

Let A, B, C, D be 98, 863, 1346, 5378 (or 98, 1346, 2018, 5378). Then A+B, A+C, A+D, B+C, B+D, and C+D are squares.

98, 99 and 100 are three consecutive integers having square factors (the second case).

(98² - 4) = (3² - 4)(6² - 4)(8² - 4),
(98² + 8) = (9² + 8)(10² + 8) = (1² + 8)(2² + 8)(9² + 8).

1² + 2² + 3² + 4² + ... + 98² = 318549, which consists of different digits.

(1 + 2 + ... + 48)(49 + 50 + ... + 63)(64 + 65 + ... + 98) = 52920²,
(1 + 2 + ... + 49)(50 + 51 + ... + 97)(98) = 20580².

(1³ + 2³ + ... + 13³)(14³ + 15³ + ... + 77³)(78³ + 79³ + ... + 98³) = 1040623584²,
(1³ + 2³ + ... + 47³)(48³ + 49³ + ... + 77³)(78³ + 79³ + ... + 98³) = 11960071200².

98⁸ = 8507630225817856, 8 + 507 + 630 + 22 + 581 + 7856 = 98²,
    8 + 507 + 630 + 225 + 8178 + 56 = 850 + 7 + 6 + 302 + 2 + 581 + 7856
  = 850 + 7 + 630 + 2 + 258 + 1 + 7856 = 850 + 7630 + 2 + 258 + 1 + 7 + 856
  = 8507 + 6 + 30 + 225 + 817 + 8 + 5 + 6 = 8507 + 630 + 225 + 8 + 178 + 56 = 98².

98² = 9604 appears in the decimal expressions of π and e:
  π = 3.14159•••9604••• (from the 33488th digit),
  e = 2.71828•••9604••• (from the 10187th digit).

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Page of Squares : First Upload March 22, 2004 ; Last Revised November 30, 2013
by Yoshio Mimura, Kobe, Japan