logo
4

The first non-trivial square : 4 = 22.

The second integer which is the sum of a square and a prime : 12 + 3.

The first integer which is the sum of 4 squares : 4 = 12 + 12 + 12 + 12.

The smallest squares containing k 4's are :
4 = 22,   144 = 122,   1444 = 382,   44944 = 2122,
6441444 = 25382,   47444544 = 68882,   4434494464 = 665922,
44424414441 = 2107712,   4243443441444 = 20599622.

The squares which begin with 4 and end in 4 are
484 = 222,   4624 = 682,   40804 = 2022,   43264 = 2082,   44944 = 2122,...

42± 3 are primes.

42 = 23 + 23.

42 = 16, the first square with an increasing sequence of digits.

42 + 52 + 62 + ... + 382 = 392 + 402 + 412 + ... + 482.

(1) x (2 + 3 + 4) = 32,   (1 + 2) x (3) x (4) = 62.

(12) x (22) x (32 + 42) = 102.

13 + 23 + 33 + 43 = (1 + 2 + 3 + 4)2,   (13 + 23 + 33) x (43) = 482.

(42 + 2) = (12 + 2 )(22 + 2 ),     (42 - 2) = (22 - 2 )(32 - 2 ),
(42 - 3) = (22 - 3 )(42 - 3 ),     (42 - 8) = (32 - 8 )(42 - 8 ).

45 = 1024, 10 + 2 + 4 = 42.

Every integer is a sum of at most 4 squares.

Every integer n is the sum of 4 nonzero squares unless
  n = 1, 2, 3, 5, 6, 8, 9, 11, 14, 17, 29, 41, 4em (m = 2, 6, 14).

4! + 1 = 52,   4! + 5! = 122.

(1/1) + (1/2) + (1/3) + (1/4) = 52/12

12 + 2 + 32 + 4 = 42

If the last three digits of a square are the same, then the digit is 4.
...0382 = ...444,   ...4622 = ...444,   ...5382 = ...444,   ...9622 = ...444

We consider the following process: Starting with an integer, form a new integer by adding the squares of its digits. Repeat these steps. Then we will obtain 1 or 4.
4 -- 42 = 16 -- 12 + 62 = 37 -- 32 + 72 = 58 -- 52 + 82 = 89 -- 82 + 92 = 145 -- 12 + 42 + 52 = 42 -- 42 + 22 = 20 -- 22 + 02 = 4.

Komachi Fractions : 42 = 45936/2871 = 73296/4581 = 98352/6147 = 60948/15237 = 68940/17235 = 69408/17352 = 86940/21735 = 94068/23517 = 94860/23715 = 150768/9423
Komachi Fractions : (4/27)2 = 7056/321489,   (4/29)2 = 9648/507123,   (4/371)2 = 720/6193845

Komachi equations:
42 = 1 + 2 + 3 + 4 + 5 * 6 - 7 - 8 - 9 = 1 + 2 + 3 - 4 * 5 + 6 + 7 + 8 + 9
  = 1 + 2 + 3 - 4 * 5 - 6 * 7 + 8 * 9, and more 256 eqiations,
42 = 9 + 8 + 7 + 6 - 5 - 4 - 3 - 2 * 1 = 9 + 8 + 7 + 6 - 5 - 4 - 3 * 2 + 1
  = 9 + 8 + 7 + 6 - 5 - 4 * 3 + 2 + 1, and more 307 eqiations,
42 = 9 + 8 + 7 + 6 + 5 + 4 - 3 - 2 * 10 = 9 + 8 + 7 + 6 + 5 - 4 - 3 - 2 - 10
  = 9 + 8 + 7 + 6 + 5 - 4 - 3 / 2 * 10, and more 239 eqiations,
42 = 12 + 22 * 342 + 52 - 672 - 82 - 92 = - 122 + 342 - 52 * 62 + 72 - 82 - 92
  = 12 + 22 + 32 * 42 + 52 + 62 - 72 - 82 - 92, and more 8 eqiations,
42 = 92 + 82 + 72 * 62 - 542 + 322 - 12 = 92 - 82 - 72 / 62 * 542 / 32 / 212
  = 92 + 82 - 72 - 62 - 52 + 42 - 32 * 22 + 12, and more 9 eqiations,
42 = 92 - 82 * 72 - 652 + 432 * 22 - 102 = 982 / 72 + 62 + 52 - 42 - 32 / 22 * 102
  = 92 - 82 + 72 + 62 + 52 - 42 + 32 - 22 - 102, and more 4 eqiations,
42 = 123 + 33 - 43 + 53 - 63 - 73 - 83 - 93 = 13 - 23 * 33 - 43 + 563 / 73 + 83 - 93
  = - 13 - 23 - 33 * 43 + 563 / 73 + 83 + 93,
42 = - 93 + 83 - 73 - 63 - 53 - 43 - 33 + 23 + 103,
42 = 94 * 84 / 74 / 64 / 544 * 34 * 214 = 984 / 74 / 64 * 544 / 34 / 214.

45 = 1024, 10 + 2 + 4 = 42,
49 = 262144, 22 + 62 + 22 + 142 + 42 = 44,
415 = 1073741824, 12 + 072 + 32 + 72 + 42 + 182 + 242 = 45.

42 = 16 appears in the decimal expressions of π and e:
  π = 3.14159•••16••• (from the 40th digit), 16 is the 2nd 2-digit square in the expression of π,
  e = 2.71828•••16••• (from the 94th digit), 16 is the 5th 2-digit square in the expression of e.


Page of Squares : First Upload August 17, 2003 ; Last Revised January 13, 2014
by Yoshio Mimura, Kobe, Japan