PARI's commands

[3.14159]

Item 16 is also the hardest to find, when dealing with larger numbers. Coincidentally, it is also the only item that is both a square and fourth power.

[3.14159]

Wow. It is barren from 1 to 80k for multiplier 120!

[3.14159]

Wait.. Prime fermat numbers are probably finite for any given base. So there are no examples for 120!. That explains it.

For item 16: There are probably no 1000+ digit examples.

Due to that, this is the only category that will be restricted to small primes, 100-750 digits.

If you find me a 1000+ digit example, the restriction will be lifted.

[CRGreathouse]

Originally Posted by 3.14159

I normally choose the multiplier and k-range.

Can you give an example for how you'd do that for #16?

[CRGreathouse]

Originally Posted by 3.14159

Item 16 is also the hardest to find, when dealing with larger numbers. Coincidentally, it is also the only item that is both a square and fourth power.

Yeah, those fourth powers sure have a lot of nonsquares.

[3.14159]

Originally Posted by CRGreathouse

Can you give an example for how you'd do that for #16?

Sure. Suppose your multiplier is 11

So (k * 11)^{(1, 2, and 4)} + 1 are all primes.

An example is (from PARI): 107251 and 11502562501 and 132308944066406250001 are primes

[3.14159]

Originally Posted by CRGreathouse

Yeah, those fourth powers sure have a lot of nonsquares.

A number can be a square and not a fourth power.

Ex: 16129 = 127².

Is 16129 a fourth power?

[science_man_88]

Originally Posted by CRGreathouse

Yeah, those fourth powers sure have a lot of nonsquares.

what he's pointing Pi out is you say both a square and a fourth power but x^4 = x^2^2 so it's a square hence the fourth power part is pointless in one sense if I read it correctly.

[3.14159]

Originally Posted by CRGreathouse

what he's pointing out is you say both a square and a fourth power but x^4 = x^2^2 so it's a square hence the fourth power part is pointless in one sense if I read it correctly.

Hence, my rebut still stands.

If it were item 25, I would only claim it is a square number.

[CRGreathouse]

Originally Posted by science_man_88

what he's pointing Pi out is you say both a square and a fourth power but x^4 = x^2^2 so it's a square hence the fourth power part is pointless in one sense if I read it correctly.

Well, the reverse rather: if you say fourth power, it's automatically a square so it's redundant to claim a number as both a square and a fourth power.

[3.14159]

Originally Posted by CRGreathouse

Well, the reverse rather: if you say fourth power, it's automatically a square so it's redundant to claim a number as both a square and a fourth power.

The set of squares that are also 4th powers are not all the squares. That clear enough for you?