Pentagonal & Triangular numbers; P_x – T_y = 1

 

A number of the form   T_n \; = \; n \,(n + 1) \,/ \,2 is called a triangular number.

The n-th pentagonal number is given by the formula   P_n \; = \; n \, (3 \, n-1) \,/ \,2
 

Let’s find pentagonal numbers   P_x   and Triangular numbers   T_y   such that

P_x \; - \; T_y \; = \; 1

x \, (3 \, x - 1)/2 \; - \; y \,(y + 1)/2 \; = \; 1

1/2 \; (3 \, x^2 \; - \; x \; - \; y^2 \; - \; y) \; = \; 1

 
Here are the first few solutions:
 

21 \; = \; T_6
22 \; = \; P_4

91 \; = \; T_{13}
92 \; = \; P_8

4186 \; = \; T_{91}
4187 \; = \; P_{53}

17766 \; = \; T_{188}
17767 \; = \; P_{109}

812175 \; = \; T_{1274}
812176 \; = \; P_{736}

3446625 \; = \; T_{2625}
3446626 \; = \; P_{1516}

157557876 \; = \; T_{17751}
157557877 \; = \; P_{10249}

668627596 \; = \; T_{36568}
668627597 \; = \; P_{21113}

30565415881 \; = \; T_{247246}
30565415882 \; = \; P_{142748}

129710307111 \; = \; T_{509333}
129710307112 \; = \; P_{294064}

5929533123150 \; = \; T_{3443699}
5929533123151 \; = \; P_{1988221}

25163130952050 \; = \; T_{7094100}
25163130952051 \; = \; P_{4095781}

1150298860475331 \; = \; T_{47964546}
1150298860475332 \; = \; P_{27692344}

4881517694390701 \; = \; T_{98808073}
4881517694390702 \; = \; P_{57046868}

223152049399091176 \; = \; T_{668059951}
223152049399091177 \; = \; P_{385704593}

946989269580844056 \; = \; T_{1376218928}
946989269580844057 \; = \; P_{794560369}

43290347284563212925 \; = \; T_{9304874774}
43290347284563212926 \; = \; P_{5372171956}

183711036780989356275 \; = \; T_{19168256925}
183711036780989356276 \; = \; P_{11066798296}

8398104221155864216386 \; = \; T_{129600186891}
8398104221155864216387 \; = \; P_{74824702789}

35638994146242354273406 \; = \; T_{266979378028}
35638994146242354273407 \; = \; P_{154140615773}

1629188928556953094766071 \; = \; T_{1805097741706}
1629188928556953094766072 \; = \; P_{1042173667088}

6913781153334235739684601 \; = \; T_{3718543035473}
6913781153334235739684602 \; = \; P_{2146901822524}

316054254035827744520401500 \; = \; T_{25141768196999}
316054254035827744520401501 \; = \; P_{14515606636441}

1341237904752695491144539300 \; = \; T_{51792623118600}
1341237904752695491144539301 \; = \; P_{29902484899561}

61312896094022025483863125041 \; = \; T_{350179657016286}
61312896094022025483863125042 \; = \; P_{202176319243084}

260193239740869591046300939711 \; = \; T_{721378180624933}
260193239740869591046300939712 \; = \; P_{416487886771328}

11894385787986237116124925856566 \; = \; T_{4877373430031011}
11894385787986237116124925856567 \; = \; P_{2815952862766733}

50476147271823947967491237764746 \; = \; T_{10047501905630468}
50476147271823947967491237764747 \; = \; P_{5800927929899029}

2307449529973235978502751753048875 \; = \; T_{67933048363417874}
2307449529973235978502751753048876 \; = \; P_{39221163759491176}

9792112377494105036102253825421125 \; = \; T_{139943648498201625}
9792112377494105036102253825421126 \; = \; P_{80796503131815076}

 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 

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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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