Curious Identities Involving Integer Squares

One characteristic feature of the identities below is the inclusion of integers with long sequences of equal successive digits. To make describing such numbers a manageable undertaking, I'll use subscripts to denote the number of repetitions of a digit, e.g., 1₅3₂ will denote 1111133, i.e., five 1s followed by two 3s. In general, d_n means a sequence of n d's.

The following identities come from the Mathematics Magazine, Vol. 35, No. 2 (March-April, 1962). For k > 1,

1_k5_k-16 = (3_k-14)², e.g., 11115556 = 3334². (Proof)
9_k-1820_k-181 = (9_k1)², e.g., 999982000081 = 999991². (Proof)
9_k-1840_k-164 = (9_k2)², e.g, 9998400064 = 99992².
9_k-1860_k-149 = (9_k3)², e.g, 9998600049 = 99993².
9_k-1880_k-136 = (9_k4)², e.g, 9998800036 = 99994².
9_k0_k25 = (9_k5)², e.g, 9999000025 = 99995².
5_k-16² - 4_k-15² = 1_2k, e.g., 556² - 445² = 111111. (Proof)

Subscripts can be easily converted to decimal notations. Observe that

1_t = 9_t / 9 = (10^t - 1) / 9.

So, for example,

3_k5_m6_n7 = 7 + 10·6·(10ⁿ - 1) / 9 + 10^{n + 1}·5·(10^m - 1) / 9 + 10^{m + n + 1}·3·(10^k - 1) / 9.

Number Curiosities

Number 8 Is Interesting
Curious Identities Involving Integer Squares
Curious Identities Involving Integer Products
Decimal Sums of Successive Integers

1_k5_k-16 = (3_k-14)².

1_k5_k-16 = 6 + 50·(10^k-1 - 1) / 9 + 10^k (10^k - 1)/9

= 6 + 5·10^k / 9 + 10^2k / 9 + 6 - 50/9 - 10^k/9

= (4·10^k + 10^2k + 4)/9

= [(10^k + 2) / 3]².

_k-1

9_k1² = (10(10^k - 1) + 1)²

= (10^k+1 - 9)²

= 10^2k+2 - 18·10^k+1 + 81

= 10^2k+2 - (20 - 2)·10^k+1 + 81

= 10^2k+2 - 2·10^k+2 + 2·10^k+1 + 81

= 10^k+2(10^k - 2) + 2·10^k+1 + 81

= 10^k+2(10(10^k-1 - 1) + 8) + 2·10^k+1 + 81

= 10^k+3(10^k-1 - 1) + 8·10^{k+2 + 2·10^k+1 + 81}

= 10^k+3(10^k-1 - 1) + 82·10^k+1 + 81

= 9_k-1820_k-181.

_k-1

_2k

5_k-16² - 4_k-15² = 1_k·10_k1

= 1_2k.

41162464

1_k5_k-16	= 6 + 50·(10^k-1 - 1) / 9 + 10^k (10^k - 1)/9
	= 6 + 5·10^k / 9 + 10^2k / 9 + 6 - 50/9 - 10^k/9
	= (4·10^k + 10^2k + 4)/9
	= [(10^k + 2) / 3]².

9_k1²	= (10(10^k - 1) + 1)²
	= (10^k+1 - 9)²
	= 10^2k+2 - 18·10^k+1 + 81
	= 10^2k+2 - (20 - 2)·10^k+1 + 81
	= 10^2k+2 - 2·10^k+2 + 2·10^k+1 + 81
	= 10^k+2(10^k - 2) + 2·10^k+1 + 81
	= 10^k+2(10(10^k-1 - 1) + 8) + 2·10^k+1 + 81
	= 10^k+3(10^k-1 - 1) + 8·10^{k+2 + 2·10^k+1 + 81}
	= 10^k+3(10^k-1 - 1) + 82·10^k+1 + 81
	= 9_k-1820_k-181.