# m^j + (m+1)^k = prime

Leroy Quet qq-quet at mindspring.com
Mon Mar 29 23:17:32 CEST 2004

```[Below posted too to sci.math]

Consider,

for m = positive integer, and letting j and k = lowest positive integers
such that

m^j +(m+1)^k = a prime  = a(m),

we have the sequence (not yet in the EIS) of primes, a(m):

(figured by hand)

3, 5, 7, 29, 11, 13, 71, 17, 19, 131, 23, 157,...

Every odd prime occurs in the sequence at least once.
(because each odd prime is of the form m +(m+1).)

But does every m lead to an odd prime??
(If not, we can just define that a(m) to be 2, where j =k =0.)

(Sequence A076846 starts the same as above, but is different.)

Another related sequence:

b(m) = least positive integer n such that
n^j + (n+1)^k = m_th odd prime, for some j and k.

b(m): 1, 2, 3, 5, 6, 8, 9, 11, 4, 15,...

Another related sequence:

c(m) = number of positive integer n's where n^j + (n+1)^k = m_th prime:

0, 1, 1, 1, 1, 1,...

(Notice I include number of ways to get 2. If j and k can be 0, however,
then we have yet another sequence.)

Another related sequence:

d(m) = lowest positive integer n where
m^n + (m+1)^n = a prime = e(m).

d(m): 1, 1, 1, 2, 1, 1, 2,...

And the primes e(m):

3, 5, 7, 41, 11, 13, 113,...

Does there exist a value for every d(m) and e(m)??

I figured all of the sequences above by hand, so I may have erred.
And I do not know which sequences above are already in the EIS, if any
are.

thanks,
Leroy Quet

```