A modified Buss conjecture

[Davide Rotondo]

Hello all, dear prime number enthusiasts. This morning I tried using the Buss conjecture, but instead of starting by taking the smallest prime greater than f(n) + 1, I considered the smallest prime that, when concatenated to f(n), yields a prime number.

I get:

1...13...3
3...37...7
21...2111...11
231...23117...17
3927...392723...23
...

I think you always get prime numbers. I think the assumptions for the Buss conjecture can be adapted for this method... what do you think?

Davide

[Davide Rotondo]

 I considered the smallest number different from 1 that, when concatenated to f(n), yields a prime number.

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[Davide Rotondo]

I have a question for you. Does this method also work for Lucky and Ludic numbers?

Davide

[M F Hasler]

On Tue, Jun 23, 2026 at 9:46 PM Davide Rotondo <david...@gmail.com> wrote:

I have a question for you. Does this method also work for Lucky and Ludic numbers?

Which method? 

On Sun, Jun 21, 2026 at 2:27 PM Davide Rotondo <david...@gmail.com> wrote:

 I considered the smallest number different from 1 that, when concatenated to f(n), yields a prime number.

OK - because it is trivial that you get primes, if you only consider primes, as you wrote first.

Yes, your condition is probably stronger than that from Buss. Both of your formulas,

Buss's B(n) = nextprime(f+1)-f  and your D(n) = min{ k>1 | isprime(concat(f,k)) }

produce a number that cannot have a common factor with f = product of earlier terms.

(Because if p divides k and f, then also concat(f,k) = f*10^n+k.)

Your formula is more restrictive, because it allows at once only k's ending in 1, 3, 7 or 9,

which BTW excludes 2 & 5, so your sequence can't have all primes as Buss's does.

(Buss's condition excludes 2 and 5 only after they have appeared.)

In both cases, a composite candidate k would have to have two or more prime factors

that did not appear earlier, which requires such candidates to be increasingly large

(at least the square of the smallest prime not yet seen),

and this lower bound is in general much larger than many primes which don't divide f.

Both of your formulas take the smallest available candidate with gcd(f,k) which 

"accidentally / by chance" yields a prime when added or concatenated to f.

Maximilian

PS:

Your sequence starts [3, 7, 11, 17, 23, 19, 13, 29, 37, 41, 67, 53, 71, 47, 97, 109, 113, 107, 31, 151, 59, 73, 127, 43, 131, 101, 137, 157, 103, 227, 149, 181, 223, 193, 211, 241, 167, 251, 79, 83, 163, 347, 197, 173, 293, 263, 367, 307, 257, 463, 383, 239, 179, 419, 571, 577, 283, 89, 373, 823, 379, 233, 541, 929, 617, 619, 631, 709, 673, 1231, 547, 199, 947, 1171, 449, 397, 311, 953, 349, 191, 409, 139, 317, 601, 431, 271, 457, 613, 509, 719, 811, 647, 659, 967]

(PARI)

f=1; for(n=1,99,forstep(k=3,oo,2, isprime(eval(Str(f,k)))&& print([n,f,k])+(f*=k)&&break)))

[Davide Rotondo]

Thank you very much Mr. Hasler

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