A257049 Integer area of integer-sided triangle such that two sides are twin primes.

6, 66, 6810, 72006, 182430, 370614, 3203694, 6353634, 28698786, 33163770, 55637466, 105470250, 151375626, 178631034, 185921166, 217064574, 376267326, 853918566, 1172755854, 1443472134, 1472632266, 2217439890, 6709586934, 13826592870, 17356640970, 18127936590

Offset: 1

Author

Michel Lagneau, Apr 23 2015

Keywords

• nonn

Comments

The area of a triangle (a,b,c) is given by Heron's formula A = sqrt(s(s-a)(s-b)(s-c)) where its side lengths are a, b, c and semiperimeter s = (a+b+c)/2.

Property of the sequence:

We observe that the sides of each triangle are of the form (k^2+2, k^2+4, 2k^2+2) and Heron's formula gives immediately the area k(2k^2+4) => a(n)= 2*A086381(n)*A253639(n).

Let the triangle (a,b,c) = (p,p+2,q) with p prime. Because q = 2t is even, Heron's formula gives the area A = sqrt((p+t+1)(p-t+1)(t-1)(t+1)). Suppose p = t+1, so p-t+1 = 2 and A = 2p*sqrt(t-1). We must have t-1 = k^2 a square, hence p=k^2+2 and q= 2t = 2(k^2+1) = 2p-2.

Consequence: the greatest prime divisor of a(n) is the length of the smallest side of the corresponding triangle if and only if p and p+2 are primes.

This statement is false if we consider a triangle of sides (p,p+2,q) where p and p+2 are composite, or p prime and p+2 composite, or p composite and p+2 prime. Example: the area of the triangle (145, 147, 194) is 10584, but the greatest prime divisor of 10584 = 2^3*3^3*7^2 is 7, and 7 is not the smallest side of the triangle, and 145 is different from 2*194-2.

The following table gives the first values (A, a, b, c) where A is the integer area, a=p, b=p+2 and c are the sides with p prime.

+---------+-------+--------+------+

| A | a=p | b= p+2 | c |

+---------+-------+--------+------+

| 6 | 3 | 5 | 4 |

| 66 | 11 | 13 | 20 |

| 6810 | 227 | 229 | 452 |

| 72006 | 1091 | 1093 | 2180 |

| 182430 | 2027 | 2029 | 4052 |

| 370614 | 3251 | 3253 | 6500 |

+---------+-------+--------+------+

Links

• Zak Seidov, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A085554, A086381, A188158, A253639.

Programs

Mathematica

nn=40000; lst={}; Do[s=(2*Prime[c]-2+Prime[c+1]+Prime[c])/2; If[IntegerQ[s], area2=s (s-2*Prime[c]+2)(s-Prime[c+1])(s-Prime[c]); If[area2>0 && IntegerQ[Sqrt[area2]] && Prime[c+1]==Prime[c]+2, AppendTo[lst, Sqrt[area2]]]], {c, nn}]; Union[lst]

Formula

a(n) = 2*A086381(n)*A253639(n). - Zak Seidov, Apr 27 2015