## Triangular numbers | T(m^2) = k^2

$T_1 \; = \; 1^2$

$T_{ \,7^2 \,} \; = \; 35^2$

I asked to solve the equation    $T_{ \,m^2 \,} \; = \; k^2$

K.D. BAJPAI,   pipo   and   Paul provided some solutions.
[see at the bottom of the page]

Now, consider the identity:
$( \,T_{2 \,n+1} \,)^2 \; + \; ( \,T_{2 \,n} \,)^2 \; = \; T_{ \,(2 \, n+1)^2 \,}$

$((2n+1)(2 \, n+2)/2)^2 + ( \,2n \,(2n+1)/2 \,)^2 = (2n+1)^2 \,( \,(2n+1)^2+1 \,)/2$

Let’s write
$( \,T_{2n+1} \,)^2 \; + \; ( \,T_{2n} \,)^2 \; = \; T_{ \,(2n+1)^2 \,} \; = \; k^2$

The integers
$a \; = \; T_{ \,2n+1 \,}$
$b \; = \; T_{ \,2n \,}$
$c \; = \; k$
form a Pythagorean triple

$(T_1)^2 + (T_0)^2 = T_{1^2} = 1^2$

$(T_7)^2 + (T_6)^2 = T_{7^2} = 35^2$
$28^2 + 21^2 = 35^2$

$(T_{41})^2 + (T_{40})^2 = T_{41^2} = 1189^2$
$861^2 + 820^2 = 1189^2$

$(T_{239})^2 + (T_{238})^2 = T_{239^2} = 40391^2$
$28680^2 + 28441^2 = 40391^2$

$(T_{1393})^2 + (T_{1392})^2 = T_{1393^2} = 1372105^2$
$970921^2 + 969528^2 = 1372105^2$

$(T_{8119})^2 + (T_{8118})^2 = T_{8119^2} = 46611179^2$
$32963140^2 + 32955021^2 = 46611179^2$

$(T_{47321})^2 + (T_{47320})^2 = T_{47321^2} = 1583407981^2$
$1119662181^2 + 1119614860^2 = 1583407981^2$

Summarizing the first few results in a table

and,

$(T_{275807})^2 + (T_{275806})^2 = T_{275807^2} = 53789260175^2$
$38034888528^2 + 38034612721^2 = 53789260175^2$

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

$(T_{1607521})^2 + (T_{1607520})^2 = T_{1607521^2} = 1827251437969^2$
$1292062686481^2 + 1292061078960^2 = 1827251437969^2$

(1292061078960,1292062686481,1827251437969)
= 1607521 × [803760,803761,1136689]

$(T_{9369319})^2 + (T_{9369318})^2 = T_{9369319^2} = 62072759630771^2$
$43892073946540^2 + 43892064577221^2 = 62072759630771^2$

(43892064577221,43892073946540,62072759630771)
= 9369319 × [4684659,4684660,6625109]

$(T_{54608393})^2 + (T_{54608392})^2 = T_{54608393^2} = 2108646576008245^2$
$1491038320325421^2+1491038265717028^2=2108646576008245^2$

(1491038265717028,1491038320325421,2108646576008245)
= 54608393 × [27304196,27304197,38613965]

$(T_{318281039})^2 + (T_{318281038})^2 = T_{318281039^2} = 71631910824649559^2$
$50651410052600280^2+50651409734319241^2=71631910824649559^2$

(50651409734319241,50651410052600280,71631910824649559)
= 318281039 × [159140519,159140520,225058681]

$(T_{1855077841})^2 + (T_{1855077840})^2 = T_{1855077841^2} = 2433376321462076761^2$
$1720656899012149561^2+1720656897157071720^2=2433376321462076761^2$

(1720656897157071720,1720656899012149561,2433376321462076761)
= 1855077841 × [927538920,927538921,1311738121]

$(T_{10812186007})^2 + (T_{10812186006})^2 = T_{10812186007^2} = 82663163018885960315^2$
$58451683130389395028^2+58451683119577209021^2=82663163018885960315^2$

(58451683119577209021,58451683130389395028,82663163018885960315)
= 10812186007 × [5406093003,5406093004,7645370045]

$(T_{63018038201})^2 + (T_{63018038200})^2 = T_{63018038201^2}$
$= 2808114166320660573949^2$

$1985636569382856677301^2+1985636569319838639100^2=2808114166320660573949^2$

(1985636569319838639100,1985636569382856677301,2808114166320660573949)
= 63018038201 × [31509019100,31509019101,44560482149]

$(T_{367296043199})^2 + (T_{367296043198})^2 = T_{367296043199^2}$
$= 95393218491883573553951^2$

$67453191675004485098400^2+67453191674637189055201^2=95393218491883573553951^2$

(67453191674637189055201,67453191675004485098400,95393218491883573553951)
= 367296043199 × [183648021599,183648021600,259717522849]

$(T_{2140758220993})^2 + (T_{2140758220992})^2 = T_{2140758220993^2}$
$= 3240561314557720840260385^2$

$2291422880375627492063521^2+2291422880373486733842528^2=3240561314557720840260385^2$

(2291422880373486733842528,2291422880375627492063521,3240561314557720840260385)
= 2140758220993 × [1070379110496,1070379110497,1513744654945]

$(T_{12477253282759})^2 + (T_{12477253282758})^2 = T_{12477253282759^2}$
$= 110083691476470624995299139^2$

$77840924741066359629967420^2+77840924741053882376684661^2=110083691476470624995299139^2$

(77840924741053882376684661,77840924741066359629967420,110083691476470624995299139)
= 12477253282759 × [6238626641379,6238626641380,8822750406821]

$(T_{72722761475561})^2 + (T_{72722761475560})^2 = T_{72722761475561^2}$
$= 3739604948885443528999910341^2$

$2644300018315705918380870141^2+2644300018315633195619394580^2=3739604948885443528999910341^2$

(2644300018315633195619394580,2644300018315705918380870141,3739604948885443528999910341)
= 72722761475561×[36361380737780,36361380737781,51422757785981]

$(T_{423859315570607})^2 + (T_{423859315570606})^2 = T_{423859315570607^2}$
$= 127036484570628609361001652455^2$

$89828359697991916746658959528^2+89828359697991492887343388921^2=127036484570628609361001652455^2$

(89828359697991492887343388921,89828359697991916746658959528,127036484570628609361001652455)
= 423859315570607 × [211929657785303,211929657785304,299713796309065]

math grad - Interest: Number theory
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### 7 Responses to Triangular numbers | T(m^2) = k^2

1. K.D. BAJPAI says:

Some more solutions are:
T41^2 = 1189^2
T239^2 = 40391^2

2. pipo says:

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

3. K.D. BAJPAI says:

The next one is found as:
T(1393^2) = 1372105^2

bajpai

4. K.D. BAJPAI says:

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

5. paul says:

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.

• benvitalis says: