(a,b,c) such that ab+1 = 2x^2, ac+1 = 2y^2, bc+1 = 2z^2

 

Find triples   (a, \; b, \; c)   such that

a \,b \; + \; 1 \; = \; 2 \,x^2
a \,c \; + \; 1 \; = \; 2 \,y^2
b \; c \; + \; 1 \; = \; 2 \,z^2

 

For example,
 

a = 49,     b = 79,     c = 943
 
(49\times 79) \; + \; 1 \; = \; 2 \; \times  \; 44^2
(49\times 943) \; + \; 1 \; = \; 2 \; \times  \; 152^2
(79\times 943) \; + \; 1 \; = \; 2 \; \times  \; 193^2
 
 
 
Find other solutions

 

Paul found:

 

ab+1 2z^2 1

 

a   =   17,   23,   31,   41

ab+1 2z^2 2

 
a   =   47,   49,   71,   73,   79,   89,   97

ab+1 2z^2 3

 
a   =   103,   113,   119,   151,   161,   199

ab+1 2z^2 4

 
a   =   353,   409,   449,   527,   679,   983,   1057,   1201,   1231,   2471,   2807

 
ab+1 2z^2 5

 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

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

3 Responses to (a,b,c) such that ab+1 = 2x^2, ac+1 = 2y^2, bc+1 = 2z^2

  1. paul says:

    Here are a few more

    7 X 103 + 1 = 2 X 19^2
    7 X 929833 + 1 = 2 X 1804^2
    103 X 929833 + 1 = 2 X 6920^2

    7 X 151 + 1 = 2 X 23^2
    7 X 5207 + 1 = 2 X 135^2
    151 X 5207 + 1 = 2 X 627^2

    7 X 193 + 1 = 2 X 26^2
    7 X 25543 + 1 = 2 X 299^2
    193 X 25543 + 1 = 2 X 1570^2

    7 X 257 + 1 = 2 X 30^2
    7 X 150593 + 1 = 2 X 726^2
    257 X 150593 + 1 = 2 X 4399^2

    7 X 257 + 1 = 2 X 30^2
    7 X 293191 + 1 = 2 X 1013^2
    257 X 293191 + 1 = 2 X 6138^2

    7 X 311 + 1 = 2 X 33^2
    7 X 21607 + 1 = 2 X 275^2
    311 X 21607 + 1 = 2 X 1833^2

    7 X 391 + 1 = 2 X 37^2
    7 X 6601 + 1 = 2 X 152^2
    391 X 6601 + 1 = 2 X 1136^2

    7 X 553 + 1 = 2 X 44^2
    7 X 115207 + 1 = 2 X 635^2
    553 X 115207 + 1 = 2 X 5644^2

    7 X 553 + 1 = 2 X 44^2
    7 X 181033 + 1 = 2 X 796^2
    553 X 181033 + 1 = 2 X 7075^2

    7 X 3031 + 1 = 2 X 103^2
    7 X 337591 + 1 = 2 X 1087^2
    3031 X 337591 + 1 = 2 X 22619^2

    7 X 4681 + 1 = 2 X 128^2
    7 X 85801 + 1 = 2 X 548^2
    4681 X 85801 + 1 = 2 X 14171^2

    7 X 11201 + 1 = 2 X 198^2
    7 X 361607 + 1 = 2 X 1125^2
    11201 X 361607 + 1 = 2 X 45002^2

    7 X 18871 + 1 = 2 X 257^2
    7 X 261671 + 1 = 2 X 957^2
    18871 X 261671 + 1 = 2 X 49689^2

    7 X 23207 + 1 = 2 X 285^2
    7 X 47561 + 1 = 2 X 408^2
    23207 X 47561 + 1 = 2 X 23492^2

    7 X 24361 + 1 = 2 X 292^2
    7 X 47561 + 1 = 2 X 408^2
    24361 X 47561 + 1 = 2 X 24069^2

    7 X 115207 + 1 = 2 X 635^2
    7 X 926743 + 1 = 2 X 1801^2
    115207 X 926743 + 1 = 2 X 231049^2

    17 X 23 + 1 = 2 X 14^2
    17 X 4247 + 1 = 2 X 190^2
    23 X 4247 + 1 = 2 X 221^2

    17 X 113 + 1 = 2 X 31^2
    17 X 8897 + 1 = 2 X 275^2
    113 X 8897 + 1 = 2 X 709^2

    17 X 497 + 1 = 2 X 65^2
    17 X 140033 + 1 = 2 X 1091^2
    497 X 140033 + 1 = 2 X 5899^2

    17 X 593 + 1 = 2 X 71^2
    17 X 28247 + 1 = 2 X 490^2
    593 X 28247 + 1 = 2 X 2894^2

    17 X 911 + 1 = 2 X 88^2
    17 X 144431 + 1 = 2 X 1108^2
    911 X 144431 + 1 = 2 X 8111^2

    17 X 1297 + 1 = 2 X 105^2
    17 X 11233 + 1 = 2 X 309^2
    1297 X 11233 + 1 = 2 X 2699^2

    17 X 4753 + 1 = 2 X 201^2
    17 X 148897 + 1 = 2 X 1125^2
    4753 X 148897 + 1 = 2 X 18811^2

    17 X 8897 + 1 = 2 X 275^2
    17 X 425153 + 1 = 2 X 1901^2
    8897 X 425153 + 1 = 2 X 43489^2

    17 X 17711 + 1 = 2 X 388^2
    17 X 51247 + 1 = 2 X 660^2
    17711 X 51247 + 1 = 2 X 21303^2

    23 X 89 + 1 = 2 X 32^2
    23 X 2009 + 1 = 2 X 152^2
    89 X 2009 + 1 = 2 X 299^2

    23 X 119 + 1 = 2 X 37^2
    23 X 3239 + 1 = 2 X 193^2
    119 X 3239 + 1 = 2 X 439^2

    23 X 75047 + 1 = 2 X 929^2
    23 X 262967 + 1 = 2 X 1739^2
    75047 X 262967 + 1 = 2 X 99335^2

    31 X 47 + 1 = 2 X 27^2
    31 X 32431 + 1 = 2 X 709^2
    47 X 32431 + 1 = 2 X 873^2

    31 X 79 + 1 = 2 X 35^2
    31 X 4879 + 1 = 2 X 275^2
    79 X 4879 + 1 = 2 X 439^2

    31 X 3151 + 1 = 2 X 221^2
    31 X 27007 + 1 = 2 X 647^2
    3151 X 27007 + 1 = 2 X 6523^2

    41 X 4361 + 1 = 2 X 299^2
    41 X 62951 + 1 = 2 X 1136^2
    4361 X 62951 + 1 = 2 X 11716^2

    47 X 233 + 1 = 2 X 74^2
    47 X 10217 + 1 = 2 X 490^2
    233 X 10217 + 1 = 2 X 1091^2

    47 X 233 + 1 = 2 X 74^2
    47 X 70703 + 1 = 2 X 1289^2
    233 X 70703 + 1 = 2 X 2870^2

    47 X 553 + 1 = 2 X 114^2
    47 X 112367 + 1 = 2 X 1625^2
    553 X 112367 + 1 = 2 X 5574^2

    47 X 623 + 1 = 2 X 121^2
    47 X 4063 + 1 = 2 X 309^2
    623 X 4063 + 1 = 2 X 1125^2

    49 X 79 + 1 = 2 X 44^2
    49 X 943 + 1 = 2 X 152^2
    79 X 943 + 1 = 2 X 193^2

    49 X 79 + 1 = 2 X 44^2
    49 X 66769 + 1 = 2 X 1279^2
    79 X 66769 + 1 = 2 X 1624^2

    71 X 617 + 1 = 2 X 148^2
    71 X 17447 + 1 = 2 X 787^2
    617 X 17447 + 1 = 2 X 2320^2

    73 X 463 + 1 = 2 X 130^2
    73 X 16969 + 1 = 2 X 787^2
    463 X 16969 + 1 = 2 X 1982^2

    73 X 9863 + 1 = 2 X 600^2
    73 X 53929 + 1 = 2 X 1403^2
    9863 X 53929 + 1 = 2 X 16308^2

    89 X 991 + 1 = 2 X 210^2
    89 X 28441 + 1 = 2 X 1125^2
    991 X 28441 + 1 = 2 X 3754^2

    89 X 3383 + 1 = 2 X 388^2
    89 X 12439 + 1 = 2 X 744^2
    3383 X 12439 + 1 = 2 X 4587^2

    113 X 1169 + 1 = 2 X 257^2
    113 X 7409 + 1 = 2 X 647^2
    1169 X 7409 + 1 = 2 X 2081^2

    119 X 569 + 1 = 2 X 184^2
    119 X 21271 + 1 = 2 X 1125^2
    569 X 21271 + 1 = 2 X 2460^2

    161 X 2471 + 1 = 2 X 446^2
    161 X 4577 + 1 = 2 X 607^2
    2471 X 4577 + 1 = 2 X 2378^2

    199 X 359 + 1 = 2 X 189^2
    199 X 1673 + 1 = 2 X 408^2
    359 X 1673 + 1 = 2 X 548^2

    199 X 1513 + 1 = 2 X 388^2
    199 X 1673 + 1 = 2 X 408^2
    1513 X 1673 + 1 = 2 X 1125^2

    199 X 3463 + 1 = 2 X 587^2
    199 X 32599 + 1 = 2 X 1801^2
    3463 X 32599 + 1 = 2 X 7513^2

    449 X 14129 + 1 = 2 X 1781^2
    449 X 14609 + 1 = 2 X 1811^2
    14129 X 14609 + 1 = 2 X 10159^2

    Paul.

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