Number Choosing Contest.

Right, now I need to figure out what rule 4 is...

39

63

95

8191

39:
Breaks Rules 2, 3, 4, 5

63:
Breaks rule 3

95:
Breaks rules 2, 3, 5

8191:
Breaks rule 5

654321

8658

654321:
Breaks rules 2, 3, 5

8658:
Breaks rules 3, 4, 5

223

239

463

911

1231

All of these break rule 5


Breaks rules 1 and 3

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SpoilerMathematica is great.

1583

2143

2239

2351

2591

All of these break rule 5

Kaynato, if I understand what you think rule 4 is correctly, you're going to have to count for quite a while to get something that is also rule 5 compliant.

Up to 33391, to be exact.

Yup, up to exactly that number!
(I'm not answering the rest because it looks like the assumptions you made were probably correct)

1: Number must not contain 7 in base ten
2: Number must be one less than, one greater than, or equal to a multiple of 7
3: Number must be prime
4: Number must contain an F when written in hexadecimal
5: Number must be expressible as the sum or difference of two nonzero cubes

Prerequisites:

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Spoiler
α: Number must be constant
β: Number must be finite
γ: Number must be real
δ: Number must be an integer
ε: Number must be positive

Looks like I made it a little easy. I'm surprised that you figured out rule 5 before rule 4. When did you figure out rule 5?