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.

 

 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

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.

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