Two consecutive integers, one is a square number, and the other triangular

To find two consecutive integers, one of which is a square number, and the other triangular:
that is,

(1)     A^2 \; - \; T_{B} \; = \; 1
(2)     T_{C} \; - \; D^2 \; = \; 1

 
(1)

1 \; = \; 2^2 \; - \; T_2
1 \; = \; 4^2 \; - \; T_5
1 \; = \; 11^2 \; - \; T_{15}
1 \; = \; 23^2 \; - \; T_{32}
1 \; = \; 64^2 \; - \; T_{90}
1 \; = \; 134^2 \; - \; T_{189}
1 \; = \; 373^2 \; - \; T_{527}
1 \; = \; 781^2 \; - \; T_{1104}
1 \; = \; 2174^2 \; - \; T_{3074}
1 \; = \; 4552^2 \; - \; T_{6437}
1 \; = \; 12671^2 \; - \; T_{17919}
1 \; = \; 26531^2 \; - \; T_{37520}
1 \; = \; 73852^2 \; - \; T_{104442}
1 \; = \; 154634^2 \; - \; T_{218685}
1 \; = \; 430441^2 \; - \; T_{608735}
1 \; = \; 901273^2 \; - \; T_{1274592}
1 \; = \; 2508794^2 \; - \; T_{3547970}
1 \; = \; 5253004^2 \; - \; T_{7428869}
1 \; = \; 14622323^2 \; - \; T_{20679087}
1 \; = \; 30616751^2 \; - \; T_{43298624}
1 \; = \; 85225144^2 \; - \; T_{120526554}
1 \; = \; 178447502^2 \; - \; T_{252362877}
1 \; = \; 496728541^2 \; - \; T_{702480239}
1 \; = \; 1040068261^2 \; - \; T_{1470878640}
1 \; = \; 2895146102^2 \; - \; T_{4094354882}
1 \; = \; 6061962064^2 \; - \; T_{8572908965}
1 \; = \; 16874148071^2 \; - \; T_{23863649055}
1 \; = \; 35331704123^2 \; - \; T_{49966575152}
1 \; = \; 98349742324^2 \; - \; T_{139087539450}
1 \; = \; 205928262674^2 \; - \; T_{291226541949}
…………………………………………………..
…………………………………………………..

 

(2)

1 \; = \; T_4 \; - \; 3^2
1 \; = \; T_{25} \; - \; 18^2
1 \; = \; T_{148} \; - \; 105^2
1 \; = \; T_{865} \; - \; 612^2
1 \; = \; T_{5044} \; - \; 3567^2
1 \; = \; T_{29401} \; - \; 20790^2
1 \; = \; T_{171364} \; - \; 121173^2
1 \; = \; T_{998785} \; - \; 706248^2
1 \; = \; T_{5821348} \; - \; 4116315^2
1 \; = \; T_{33929305} \; - \; 23991642^2
1 \; = \; T_{197754484} \; - \; 139833537^2
1 \; = \; T_{1152597601} \; - \; 815009580^2
1 \; = \; T_{6717831124} \; - \; 4750223943^2
1 \; = \; T_{39154389145} \; - \; 27686334078^2
1 \; = \; T_{228208503748} \; - \; 161367780525^2
………………………………..
………………………………..

 
 
 
 

 
 
 
 
 
 
 

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About benvitalis

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

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