A245648 The largest member 'c' of the Pythagorean triples (a,b,c) ordered by increasing c, where the triples consist of a triangular number, a square number and a pentagonal number.

5, 15, 145, 2775

Offset: 1

Author

Ivan N. Ianakiev, Jul 28 2014

Keywords

• nonn
• more
• hard

Comments

Next term comes from a triple with c > 10^5.

From Michel Marcus, Apr 08 2021: (Start)

The 4 known triples that satisfy the requisite are [3,4,5], [9,12,15], [100, 105, 145], [900, 2625, 2775].

Let po(n) be A176774(n), the least polygonality of a number.

po([3,4,5]) = [3,4,5]; <-----

po([9,12,15]) = [4,5,3];

po([100,105,145]) = [4,3,5]; <-----

po([900,2625,2775]) = [4,5,3].

So for the 2 highlighted triples, we have a-gonal^2 + b-gonal^2 = c-gonal^2. Are there other Pythagorean triples with the same property?

Let nb(n) be A177025(n) is the number of ways to represent n as a polygonal number.

nb([3,4,5]) = [1,1,1]; <-----

nb([9,12,15]) = [4,5,3];

nb([100,105,145]) = [4,3,5];

nb([900,2625,2775]) = [4,5,3].

So for the highlighted triple, we get [1,1,1]. Are there other Pythagorean triples with the same property? (End)

Regarding the first question by Michel Marcus, if such triple [x,y,z] exists, then z > 10^4. Regarding his second question, if such triple exists, then z > 10^7. - Ivan N. Ianakiev, Dec 16 2021

a(5) > 10^11, if it exists. - Giovanni Resta, Apr 15 2021

Examples

			a(1) = 5 as the first such Pythagorean triple is (3,4,5). The next three triples are (9,12,15), (100,105,145), (900,2625,2775).
		

Crossrefs

Cf. A000217, A000290, A000326, A245646, A245647.

Cf. A176774, A177025.

Programs

Mathematica

n=10^3;ppt={};list={};pos=1;t[x_]:=(IntegerPart[Sqrt[2*x]])*(IntegerPart[Sqrt[2*x]]+1)/2;ls[x_]:=Length[Sqrt[x]];lis[x_]:=Length[IntegerPart[Sqrt[x]]];lp[x_]:=Length[(Sqrt[24*x+1]+1)/6];lip[x_]:=Length[IntegerPart[(Sqrt[24*x+1]+1)/6]];Do[y=x+1;z=y+1;While[z<=n,While[z^2