I asked to solve the equation

K.D. BAJPAI, pipo and Paul provided some solutions.

[see at the bottom of the page]

Now, consider the identity:

Let’s write

The integers

form a Pythagorean triple

Summarizing the first few results in a table

and,

(38034612721,38034888528,53789260175) = 275807 × [137903,137904,195025]

(1292061078960,1292062686481,1827251437969)

= 1607521 × [803760,803761,1136689]

(43892064577221,43892073946540,62072759630771)

= 9369319 × [4684659,4684660,6625109]

(1491038265717028,1491038320325421,2108646576008245)

= 54608393 × [27304196,27304197,38613965]

(50651409734319241,50651410052600280,71631910824649559)

= 318281039 × [159140519,159140520,225058681]

(1720656897157071720,1720656899012149561,2433376321462076761)

= 1855077841 × [927538920,927538921,1311738121]

(58451683119577209021,58451683130389395028,82663163018885960315)

= 10812186007 × [5406093003,5406093004,7645370045]

(1985636569319838639100,1985636569382856677301,2808114166320660573949)

= 63018038201 × [31509019100,31509019101,44560482149]

(67453191674637189055201,67453191675004485098400,95393218491883573553951)

= 367296043199 × [183648021599,183648021600,259717522849]

(2291422880373486733842528,2291422880375627492063521,3240561314557720840260385)

= 2140758220993 × [1070379110496,1070379110497,1513744654945]

(77840924741053882376684661,77840924741066359629967420,110083691476470624995299139)

= 12477253282759 × [6238626641379,6238626641380,8822750406821]

(2644300018315633195619394580,2644300018315705918380870141,3739604948885443528999910341)

= 72722761475561×[36361380737780,36361380737781,51422757785981]

(89828359697991492887343388921,89828359697991916746658959528,127036484570628609361001652455)

= 423859315570607 × [211929657785303,211929657785304,299713796309065]

Some more solutions are:

T41^2 = 1189^2

T239^2 = 40391^2

T(41^2) = 1189^2

T(239^2) = 40391^2.

Recurrence for the n: n(p) = 6*n(p-1) – n(p-2)

Recurrence for the k: k(p) = 34*k(p-1) – k(p-2)

pipo

The next one is found as:

T(1393^2) = 1372105^2

bajpai

No more results for n <= 5*10^6.

bajpai

Here are the first 20

T 1^2 = 1^2

T 7^2 = 35^2

T 41^2 = 1189^2

T 239^2 = 40391^2

T 1393^2 = 1372105^2

T 8119^2 = 46611179^2

T 47321^2 = 1583407981^2

T 275807^2 = 53789260175^2

T 1607521^2 = 1827251437969^2

T 9369319^2 = 62072759630771^2

T 54608393^2 = 2108646576008245^2

T 318281039^2 = 71631910824649559^2

T 1855077841^2 = 2433376321462076761^2

T 10812186007^2 = 82663163018885960315^2

T 63018038201^2 = 2808114166320660573949^2

T 367296043199^2 = 95393218491883573553951^2

T 2140758220993^2 = 3240561314557720840260385^2

T 12477253282759^2 = 110083691476470624995299139^2

T 72722761475561^2 = 3739604948885443528999910341^2

T 423859315570607^2 = 127036484570628609361001652455^2

Paul.

Thank you all for your answers, Please check out my post.

I posted the rest of your results