**Information updated on 22 February 2023.**

When the Ceiling Square c^{2} for a positive odd integer m leaves a remainder r that is not a perfect square, that r still must be congruent to 0 or 1 modulo 4, corresponding to c being odd or even.

I have started exploring how odd positive integers distribute themselves among numbers of the form x^{2}-r where r is not a perfect square, and the above fact tells me that when I’m looking at how these distributions relate to Ceiling Squares and their remainders, I can narrow my scope to the remainders that are not perfect squares and are either multiples of 4 or integers 1 greater than a multiple of 4. Also, since we want odd integers, the Ceiling Square will be odd or even as the remainder is even or odd, respectively. That also thins out, a little bit, the candidate numbers we will observe.

So let’s start looking, shall we? Welcome to my scratchpad (which is whatever I have handy that suits the purpose).

```
No integers satisfy for remainders between 1 and 4.
Between 4 and 9:
c^2-5, starting at c=4:
4^2-5 = 11, prime.
6^2-5 = 31, prime.
8^2-5 = 59, prime.
10^2-5 = 95 = 5*19 = 12^2-7^2. (Ascent 2)
12^2-5 = 139, prime.
14^2-5 = 191, prime.
16^2-5 = 251, prime.
18^2-5 = 319 = 11*29 = 20^2-9^2. (Ascent 2)
20^2-5 = 395 = 5*79 = 42^2-37^2. (Ascent 22)
22^2-5 = 479, prime.
And so on.
c^2-8, starting at c=5:
5^2-8 = 17, prime.
7^2-8 = 41, prime.
9^2-8 = 73, prime.
11^2-8 = 113, prime.
13^2-8 = 161 = 7*23 = 15^2-8^2. (Ascent 2)
15^2-8 = 217 = 7*31 = 19^2-12^2. (Ascent 4)
17^2-8 = 281, prime.
19^2-8 = 353, prime.
21^2-8 = 433, prime.
23^2-8 = 521, prime.
And so on.
Between 9 and 16:
c^2-12, starting at c=7:
7^2-12 = 37, prime.
9^2-12 = 69 = 3*23 = 13^2-10^2. (Ascent 4)
11^2-12 = 109, prime.
13^2-12 = 157, prime.
15^2-12 = 213 = 3*71 = 37^2-34^2. (Ascent 22)
17^2-12 = 277, prime.
19^2-12 = 349, prime.
21^2-12 = 429 = (among others) 13*33. (Ascent 2)
23^2-12 = 517 = 11*47 = 29^2-18^2. (Ascent 6)
25^2-12 = 613, prime.
c^2-13, starting at c=8:
8^2-13 = 51 = 3*17 = 10^2-7^2. (Ascent 2)
10^2-13 = 87 = 3*29 = 16^2-13^2. (Ascent 6)
12^2-13 = 131, prime.
14^2-13 = 183 = 3*61 = 32^2-29^2. (Ascent 18)
16^2-13 = 243 = (among others) 9*27 = 18^2-9^2. (Ascent 2)
18^2-13 = 311, prime.
20^2-13 = 387 = (among others) 9*43 = 26^2-17^2. (Ascent 6)
22^2-13 = 471 = 3*157 = 80^2-77^2. (Ascent 58)
24^2-13 = 563, prime.
26^2-13 = 663 = (among others) 17*39 = 28^2-11^2. (Ascent 2)
Between 16 and 25:
c^2-17, starting at c=10:
10^2-17 = 83, prime.
12^2-17 = 127, prime.
14^2-17 = 179, prime.
16^2-17 = 239, prime.
18^2-17 = 307, prime.
20^2-17 = 383, prime.
22^2-17 = 467, prime.
24^2-17 = 559 = 13*43 = 28^2-15^2. (Ascent 4)
26^2-17 = 659, prime.
28^2-17 = 767 = 13*59 = 36^2-23^2. (Ascent 8)
c^2-20, starting at c=11:
11^2-20 = 101, prime.
13^2-20 = 149, prime.
15^2-20 = 205 = 5*41 = 23^2-18^2. (Ascent 8)
17^2-20 = 269, prime.
19^2-20 = 341 = 11*31 = 21^2-10^2. (Ascent 2)
21^2-20 = 421, prime.
23^2-20 = 509, prime.
25^2-20 = 605 = 5*11^2 = (among others) 33^2-22^2. (Ascent 8)
27^2-20 = 709, prime.
29^2-20 = 821, prime.
c^2-21, starting at c=12:
12^2-21 = 123 = 3*41 = 22^2 - 19^2. (Ascent 10)
14^2-21 = 175 = 5^2*7 = (among others) 16^2-9^2. (Ascent 2)
m=pqr = [(p+qr)/2]^2-[(p-qr)/2]^2
= [(pq+r)/2)^2-[(pq-r)/2]^2.
5^2-2^2 = 21 = 3*7, ascent 0.
6^2-1^2 = 35 = 5*7, ascent 0.
7^2-4^2 = 33 = 3*11, ascent 0.
7^2-2^2 = 45 = 5*9, ascent 0.
8^2-5^2 = 39 = 3*13, ascent 0.
8^2-3^2 = 55 = 5*11, ascent 0.
8^2-1^2 = 63 = 7*9, ascent 0.
9^2-4^2 = 65 = 5*13, ascent 0.
9^2-2^2 = 77 = 7*11, ascent 0.
10^2-7^2 = 51 = 3*17, ascent 2, remainder 13.
10^2-3^2 = 91 = 7*13, ascent 0.
10^2-1^2 = 99 = 9*11, ascent 0.
11^2-8^2 = 57 = 3*19, ascent 2, remainder 24.
11^2-6^2 = 85 = 5*17, ascent 0.
11^2-4^2 = 105 = 7*15, ascent 0.
11^2-2^2 = 117 = 9*13, ascent 0.
12^2-7^2 = 95 = 5*19, ascent 2, remainder 5.
12^2-5^2 = 119 = 7*17, ascent 0.
12^2-1^2 = 143 = 11*13, ascent 0.
13^2-10^2 = 69 = 3*23, ascent 4, remainder 12.
13^2-8^2 = 105 = 7*15, ascent 0.
13^2-6^2 = 133 = 7*19, ascent 0.
13^2-4^2 = 153 = 9*17, ascent 0.
13^2-2^2 = 165 = 11*15, ascent 0.
14^2-11^2 = 75 = 5*15, ascent 0.
14^2-9^2 = 115 = 5*23, ascent 2, remainder 29.
14^2-5^2 = 171 = 9*19, ascent 0.
14^2-3^2 = 187 = 11*17, ascent 0.
14^2-1^2 = 195 = 13*15, ascent 0.
15^2-8^2 = 161 = 7*23, ascent 2, remainder 8.
15^2-4^2 = 209 = 11*19, ascent 0.
15^2-2^2 = 221 = 13*17, ascent 0.
16^2-13^2 = 87 = 3*29, ascent 6, remainder 13.
16^2-11^2 = 135 = 9*15, ascent 0.
16^2-9^2 = 175 = 7*25, ascent 2, remainder 21.
16^2-7^2 = 207 = 9*23, ascent 0.
16^2-5^2 = 231 = 11*21, ascent 0.
16^2-3^2 = 247 = 13*19, ascent 0.
16^2-1^2 = 255 = 15*17, ascent 0.
17^2-14^2 = 93 = 3*31, ascent 6, remainder 28.
17^2-12^2 = 145 = 5*29, ascent 4, remainder 24.
17^2-10^2 = 189 = 9*21, ascent 0.
17^2-8^2 = 225 = 15*15, ascent 0.
17^2-6^2 = 253 = 11*23, ascent 0.
17^2-4^2 = 273 = 13*21, ascent 0.
17^2-2^2 = 285 = 15*19, ascent 0.
18^2-13^2 = 155 = 5*31, ascent 4, remainder 41.
18^2-11^2 = 203 = 7*29, ascent 2, remainder 53.
18^2-7^2 = 275 = 11*25, ascent 0.
18^2-5^2 = 299 = 13*23, ascent 0.
18^2-1^2 = 323 = 17*19, ascent 0.
```

I have updated this with additional entries, as well as some newer observations, and will continue filling out my own copy of this list of values and properties.

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