## Concatenations: (A,B); A||B is a square and B||A a prime number

Consider the pairs   (31, 36)   and   (229, 441)

$31$   is a prime number.
$36 = 6^2$   is a perfect square.

$36 \; || \; 31 \; = \; 3631$   is a prime number.

$31 \; || \; 36 \; = \; 3136 \; = \; 56^2$   is a perfect square.

$441 = 21^2$   is a perfect square.
$229$   is a prime number.

$441 \; || \; 229 \; = \; 441229$   is a prime number.

$229 \; || \; 441 \; = \; 229441 \; = \; 479^2$   is a perfect square.

Find other pairs   $(A, \; B)$   such that

$A \; || \; B$   is a prime number, and   $B \; || \; A$   is a perfect square.

where, $A$   is a prime number, and   $B$   is a perfect square.

Paul’s solutions:

(3,   30625),   (3,   330625),   (3,   422500)
$3$   is a prime number.
$30625 \; = \; 175^2$   ……….   $330625 \; = \; 575^2$   ……….   $422500 \; = \; 650^2$

$3 \; || \; 30625 \; = \; 330625 \; = \; 575^2$
$30625 \; || \; 3 \; = \; 306253$   is a prime number.

$3 \; || \; 330625 \; = \; 3330625 \; = \; 1825^2$
$330625 \; || \; 3 \; = \; 3306253$   is a prime number.

$3 \; || \; 422500 \; = \; 3422500 \; = \; 1850^2$
$422500 \; || \; 3 \; = \; 4225003$   is a prime number.

(13,   22500)
$13$   is a prime number   ……….   $22500 \; = \; 150^2$
$13 \; || \; 22500 = 1322500 \; = \; 1150^2$
$22500 \; || \; 13 \; = \; 2250013$   is a prime number.

(109,   2025)
$109$   is a prime number   ……….   $2025 \; = \; 45^2$
$109 \; || \; 2025 \; = \; 1092025 \; = \; 1045^2$
$2025 \; || \; 109 \; = \; 2025109$   is a prime number.

(229,   44100)
$229$   is a prime number   ……….   $44100 \; = \; 210^2$
$229 \; || \; 44100 \; = \; 22944100 \; = \; 4790^2$
$44100 \; || \; 229 \; = \; 44100229$   is a prime number.

(421,   59049)
$421$   is a prime number   ……….   $59049 \; = \; 243^2$
$421 \; || \; 59049 \; = \; 42159049 \; = \; 6493^2$
$59049 \; || \; 421 \; = \; 59049421$   is a prime number.

(433,   680625)
$433$   is a prime number   ……….   $680625 \; = \; 825^2$
$433 \; || \; 680625 \; = \; 433680625 \; = \; 20825^2$
$680625 \; || \; 433 \; = \; 680625433$   is a prime number.

(577,   921600)
$577$   is a prime number   ……….   $921600 \; = \; 960^2$
$577 \; || \; 921600 \; = \; 577921600 \; = \; 24040^2$
$921600 \; || \; 577 \; = \; 921600577$   is a prime number.

(601,   230400)
$601$   is a prime number   ……….   $230400 \; = \; 480^2$
$601 \; || \; 230400 \; = \; 601230400 \; = \; 24520^2$
$230400 \; || \; 601 \; = \; 230400601$   is a prime number.

(937,   400689)
$937$   is a prime number   ……….   $400689 \; = \; 633^2$
$937 \; || \; 400689 \; = \; 937400689 \; = \; 30617^2$
$400689 \; || \; 937 \; = \; 400689937$   is a prime number.

(997,   106929)
$997$   is a prime number   ……….   $106929 \; = \; 327^2$
$997 \; || \; 106929 \; = \; 997106929 \; = \; 31577^2$
$106929 \; || \; 997 \; = \; 106929997$   is a prime number.

(1009,   269361)
$1009$   is a prime number   ……….   $269361 \; = \; 519^2$
$1009 \; || \; 269361 \; = \; 1009269361 \; = \; 31769^2$
$269361 \; || \; 1009 \; = \; 2693611009$   is a prime number.

(1063,   99225)
$1063$   is a prime number   ……….   $99225 \; = \; 315^2$
$1063 \; || \; 99225 \; = \; 106399225 \; = \; 10315^2$
$99225 \; || \; 1063 \; = \; 992251063$   is a prime number.

(1531,   15876)
$1531$   is a prime number   ……….   $15876 \; = \; 126^2$
$1531 \; || \; 15876 \; = \; 153115876 \; = \; 12374^2$
$15876 \; || \; 1531 \; = \; 158761531$   is a prime number.

(2437,   3969)
$2437$   is a prime number   ……….   $3969 \; = \; 63^2$
$2437 \; || \; 3969 \; = \; 24373969 \; = \; 4937^2$
$3969 \; || \; 2437 \; = \; 39692437$   is a prime number.

(2551,   2601)
$2551$   is a prime number   ……….   $2601 \; = \; 51^2$
$2551 \; || \; 2601 \; = \; 5051^2$
$2601 \; || \; 2551 \; = \; 26012551$   is a prime number.

(3847,   224676)
$3847$   is a prime number   ……….   $224676 \; = \; 474^2$
$3847 \; || \; 224676 \; = \; 3847224676 \; = \; 62026^2$
$224676 \; || \; 3847 \; = \; 2246763847$   is a prime number.

(6133,   54756)
$6133$   is a prime number   ……….   $54756 \; = \; 234^2$
$6133 \; || \; 54756 \; = \; 613354756 \; = \; 24766^2$
$54756 \; || \; 6133 \; = \; 547566133$   is a prime number.

math grad - Interest: Number theory
This entry was posted in Number Puzzles and tagged . Bookmark the permalink.

### 2 Responses to Concatenations: (A,B); A||B is a square and B||A a prime number

1. paul says:

Here are those where a and b <=1000, (the concatenation symbol may not print)

3|| 30625 = 330625 Is a Square
30625||3 = 306253 Is a Prime

3|| 330625 = 3330625 Is a Square
330625||3 = 3306253 Is a Prime

3|| 422500 = 3422500 Is a Square
422500||3 = 4225003 Is a Prime

13|| 22500 = 1322500 Is a Square
22500||13 = 2250013 Is a Prime

31|| 36 = 3136 Is a Square
36||31 = 3631 Is a Prime

109|| 2025 = 1092025 Is a Square
2025||109 = 2025109 Is a Prime

229|| 441 = 229441 Is a Square
441||229 = 441229 Is a Prime

229|| 44100 = 22944100 Is a Square
44100||229 = 44100229 Is a Prime

421|| 59049 = 42159049 Is a Square
59049||421 = 59049421 Is a Prime

433|| 680625 = 433680625 Is a Square
680625||433 = 680625433 Is a Prime

577|| 921600 = 577921600 Is a Square
921600||577 = 921600577 Is a Prime

601|| 230400 = 601230400 Is a Square
230400||601 = 230400601 Is a Prime

937|| 400689 = 937400689 Is a Square
400689||937 = 400689937 Is a Prime

997|| 106929 = 997106929 Is a Square
106929||997 = 106929997 Is a Prime

1009|| 269361 = 1009269361 Is a Square
269361||1009 = 2693611009 Is a Prime

1063|| 99225 = 106399225 Is a Square
99225||1063 = 992251063 Is a Prime

1531|| 15876 = 153115876 Is a Square
15876||1531 = 158761531 Is a Prime

2437|| 3969 = 24373969 Is a Square
3969||2437 = 39692437 Is a Prime

2551|| 2601 = 25512601 Is a Square
2601||2551 = 26012551 Is a Prime

3847|| 224676 = 3847224676 Is a Square
224676||3847 = 2246763847 Is a Prime

6133|| 54756 = 613354756 Is a Square
54756||6133 = 547566133 Is a Prime

Paul.

• benvitalis says:

Nice work! I posted your solutions