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

f.firoozbakht at sci.ui.ac.ir f.firoozbakht at sci.ui.ac.ir
Tue Mar 30 05:55:03 CEST 2004

```On Mon, 29 Mar 2004, Leroy Quet wrote:
...
>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,...
>...
> 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,...
> ...

Dear Leroy,

The " lowest positive integers such that m^j+(m+1)^k=a prime =a(m) "
isn't well defined.
For example take m=12, 12^1+13^2 and 12^2+13^1 are both primes now
by your definition we don't know a(12)=12^1+13^2 or a(12)=12^2+13^1,
You wrote " 157 "or 12^2+13^1 as a(12),I think it is better that
a(m) is defined :
"smallest prime of the for m^j+(m+1)^k where j and k are positive
integers.

Note that:
1^1+2^2=2-th odd prime so b(2)=1  ( b(2)!= 2 )
2^2+3^1=3-th odd prime so b(3)=2  ( b(3)!= 3 )
2^1+3^2=4-th odd prime so b(4)=2  ( b(4)!= 5 )
3^2+4^1=5-th odd prime so b(5)=3  ( b(5)!= 6 )
3^1+4^2=7-th odd prime so b(7)=3  ( b(7)!= 9 )
5^2+6^1=10-th odd prime so b(10)=5 ( b(10)!= 15 )

So b(m) for m=1,2,...,10:  1,1,2,2,3,8,3,11,4,5

I get b(m) for m<=20 : 1,1,2,2,3,8,3,11,4,5,18,4,6,23,26,29,5,33,7,8
,this is handwork (checks are needed).

b(m) isn't in EIS,d(m) is in EIS d(m)=A057856(m)
d(m) for m=1,2,...,45 (except for m=28):

1,1,1,2,1,1,2,1,1,32,1,2,4,1,1,4,2,1,2,1,1,2,1,
2,2,1,4,??,1,1,4,2,1,2,1,1,4,4,1,4,1,2,4,1,16

d(28)(if there exist)is greater than 8191.

It is obvious that d(m) is of the form 2^k(m),the sequence k(m)
is also in EIS ,k(m)=A058064(m), the sequence e(m) isn't in EIS.
e(m) for m=1,2,...,45 (except for m=28):

3,5,7,41,11,13,113,17,19,2211377674535255285545615254209921,
23,313,66977,29,31,149057,613,37,761,41,43,1013,47,1201,1301,
53,1146097,??,59,61,1972097,2113,67,2381,71,73,3959297,4398577,
79,5385761,83,3613,7166897,89,684655198104511486296198721

Regards,

Farideh

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