The name "near-square number" is used here for a number of the form 
, where 
is a positive integer
and 
is an integer. A near-square prime is a prime
number of such a form for a specified value of 
.
Numbers of the form 
for 
, 2, ... are 
, 
, 4, 11, 20, 31, 44, 59, 76, 95, ... (OEIS A028875).
These are prime for indices 
, 6, 8, 12, 14, 16, ... (OEIS A028876),
corresponding to the primes 11, 31, 59, 139, 191,
251, 479, ... (OEIS A028877).
Numbers of the form 
for 
, 2, ... are 
, 0, 5, 12, 21, 32, 45, 60, ...
(OEIS A028347). Since they can always be factored
as 
for 
, the only (positive) prime
of this form is 5.
Numbers of the form 
for 
, 2, ... are 
, 1, 6, 13, 22, 33, 46, 61, ...
(OEIS A028872). These are prime
for indices 
, 8, 10, 14, 20, 26, 32, ... (OEIS A028873),
corresponding to the primes 13, 61, 97, 193, 397,
673, ... (OEIS A028874).
Numbers of the form 
for 
, 2, ... are 
, 2, 7, 14, 23, 34, 47, 62, ...
(OEIS A008865). These are prime
for indices 
, 3, 5, 7, 9, 13, 15, 19, 21, 27, ... (OEIS A028870),
corresponding to the primes 2, 7, 23, 47, 79, 167,
223, 359, 439, ... (OEIS A028871). The "Kynea
primes" (see below) are special cases of this
form.
Numbers of the form 
for 
, 2, ... are 0, 3, 8, 15, 24, 35, 48, 63, ... (OEIS
A005563). Since they can always be factored
as 
for 
, the only prime
of this form is 3.
Numbers of the form 
for 
, 2, ... are 2, 5, 10, 17, 26, 37, 50, ... (OEIS A002522). These are prime
for indices 
, 2, 4, 6, 10, 14, 16, 20, 24, 26, 36, ... (OEIS A005574), corresponding to the primes
2, 5, 17, 37, 101, 197, 257, 401, 577, 677, ... (OEIS A002496).
Fermat primes other than 3 and prime generalized Fermat numbers of the form 
with 
are of this form. For example, the proven prime 
has 13,426,224 decimal digits (https://t5k.org/primes/page.php?id=141162).
Numbers of the form 
for 
, 2, ... are 3, 6, 11, 18, 27, 38, 51, 66, 83, 102,
... (OEIS A059100). These are prime
for indices 
, 3, 9, 15, 21, 33, 39, 45, 57, 81, 99, ... (OEIS
A067201), corresponding to the primes
3, 11, 83, 227, 443, 1091, 1523, 2027, ... (OEIS A056899).
Numbers of the form 
for 
, 2, ... are 4, 7, 12, 19, 28, 39, 52, 67, ... (OEIS
A117950). These are prime
for indices 
, 4, 8, 10, 14, 22, 28, 38, 50, 52, 62, ... (OEIS
A049422), corresponding to the primes
7, 19, 67, 103, 199, 487, 787, 1447, ... (OEIS A049423).
Numbers of the form 
for 
, 2, ... are 5, 8, 13, 20, 29, 40, 53, 68, 85, ...
(OEIS A087475). These are prime
for indices 
, 3, 5, 7, 13, 15, 17, 27, ... (OEIS A007591),
corresponding to the primes 5, 13, 29, 53, 173, 229,
293, 733, ... (OEIS A005473).
Numbers of the form 
for 
, 2, ... are 6, 9, 14, 21, 30, 41, 54, 69, ... (OEIS
A117951). These are prime
for indices 
, 12, 36, 48, 72, 78, 96, ... (OEIS A078402),
corresponding to the primes 41, 149, 1301, 2309,
5189, ... (OEIS A056905).
For 
, 2, ..., the first few near-square numbers of the
form 
are 
, 7, 47, 223, 959, 3967, ... (OEIS A093112).
As of Jun. 2016, a total of 42 primes of this
form (arbitrarily dubbed Carol primes by their original
investigator in reference to a personal acquaintance) are known. The first few have
indices 2, 3, 4, 6, 7, 10, 12, 15, 18, 19, 21, 25, 27, 55, 129, ... (OEIS A091515),
corresponding to the primes 7, 47, 223, 3967, 16127,
1046527, ... (OEIS A091516). Some large
primes of this form are summarized in the following table.
 | decimal digits | discoverer |
 | 152916 | C. Emmanuel (May 7, 2007; Harvey; https://t5k.org/primes/page.php?id=80384) |
 | 313290 | M. Rodenkirch (Apr. 14, 2016) |
 | 393441 | M. Rodenkirch (Jun. 15, 2016) |
 | 414243 | M. Rodenkirch (reported Jul. 2016; https://t5k.org/primes/page.php?id=121867) |
 | 418812 | M. Rodenkirch (reported Jul. 2016; https://t5k.org/primes/page.php?id=121905) |
For 
, 1, ..., the first few near-square numbers of the
form 
are 2, 7, 23, 79, 287, 1087, 4223, 16639, ... (OEIS A093069
begins at 
). As of Jun. 2016, a total of 50 primes
of this form (arbitrarily dubbed Kynea primes by
their original investigator in reference to a personal acquaintance) are known. The
first few have indices 0, 1, 2, 3, 5, 8, 9, 12, 15, 17, 18, 21, ... (OEIS A091513)
and are given by 2, 7, 23, 79, 1087, 66047, 263167, 16785407, ... (OEIS A091514).
Some large primes of this form are summarized in the
following table.
 | decimal digits | discoverer |
 | 169553 | C. Emmanuel (October 2005); https://t5k.org/primes/page.php?id=75878 |
 | 222510 | M. Rodenkirch (Feb. 2016) |
 | 226405 | M. Rodenkirch (Feb. 7, 2016) |
 | 266142 | M. Rodenkirch (Feb. 28, 2016) |
 | 374146 | M. Rodenkirch (May 30, 2016) |
 | 398250 | M. Rodenkirch (Jun. 19, 2016) |
 | 513419 | R. Propper (reported Jul. 2019; https://t5k.org/primes/page.php?id=126646) |
." No Prime Left Behind. Updated Oct. 23,
2025. 
-Tuplets.
"Exhaustive Search Table for Generalized (Carol-, Kynea-, Repunit-, Wagstaff-,
Mersenne) Primes." Updated Oct. 15, 2026.