A237518 Least primes that together with prime(n) forms a Heronian triangle, starting at n = 2.
5, 3, 4729, 13, 5, 17, 37, 5280071830550089, 5, 97, 13, 17, 61, 1824001, 53, 109, 11, 3301, 1009, 19, 241, 241, 17, 11, 29, 409, 6841, 11, 17, 3169, 181, 41, 157, 3, 457, 13, 10369, 231781748893580717709514473745694370721, 173, 277, 19, 7297, 31, 53, 3049, 373
Offset: 2
Keywords
Examples
a(18)=11 as prime(18)=61, the triple (11, 60, 61) is Heronian and right angled with area=330 and 61 is the least such prime. prime(18)=61==1 mod 4 and a(18)=11==3 mod 4 and prime(18)>a(18).
Links
- Noam Elkies, Heronian triangles with two sides that are prime, Mathoverflow, 2013.
- Eugen J. Ionascu, Florian Luca, Pantelimon Stanica, Heron triangles with two fixed sides, arXiv:math/0608185 [math.NT], 2006.
- Eugen J. Ionascu, Florian Luca, Pantelimon Stanica, Heron triangles with two fixed sides, Journal of Number Theory, Volume 126, Issue 1, September 2007, Pages 52-67.
- Pantelimon Stanica, Santanu Sarkar, Sourav Sen Gupta, Subhamoy Maitra, and Nirupam Kar, Counting Heron triangles with Constraints, Integers, Vol. 13, 2013.
Programs
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Mathematica
maxn = 150000; nn=Prime[Range[maxn]]; lst={}; nn1=Prime[Range[2, 100]]; Do[Do[s=(a+b+c)/2; If[IntegerQ[s], area2=s(s-a)(s-b)(s-c); If[area2>0 && IntegerQ[Sqrt[area2]], (AppendTo[lst, b]; Break[])]]; If[b==Prime[maxn], AppendTo[lst, 0]; Break[]], {b, nn}, {a, b-c+2, b+c-2, 2}], {c, nn1}]; lst (* 1st Program *) q=23; d=1; nextpair[{y0_, x0_}] := (y=23; x=4; y1=y*y0+x*x0*33; x1=x0*y+y0*x; {y1, x1}); pair=nextpair[{0, q}]; While[!PrimeQ[(pair[[2]]-d)/2] && !PrimeQ[(pair[[2]]-d)/2+d], pair=nextpair[pair]]; primepair={(pair[[2]]-d)/2, (pair[[2]]-d)/2+d}; primepair(* 2nd Program *) q=167; d=25; y=88751; x=2150; nextpair[{y0_, x0_}] := (If[IntegerQ[(q^2-d^2)/16], k=(q^2-d^2)/16, k=(q^2-d^2)/4]; y1=y*y0+x*x0*k; x1=x0*y+y0*x; {y1, x1}); pair=nextpair[{0, q}]; While[!PrimeQ[(pair[[2]]-d)/2] && !PrimeQ[(pair[[2]]+d)/2], pair=nextpair[pair]]; primepair={(pair[[2]]-d)/2, (pair[[2]]+d)/2}; primepair(* 3rd Program *)
Formula
Apart from searching through the first 150000 prime numbers for each prime(n) to form a Heronian triangle (1st Mathematica program), more difficult primes e.g. prime(9)=23 and prime(39)=167 require Pell-type equations to be solved and searched for these least primes (2nd and 3rd Mathematica programs). If a Heronian triangle has side length triples of the form (q, p, p+d) where q = prime(n) and d is odd such that 0 > d > p, then the Pell-type equation is of the form Y^2 - K*X^2 = -J with Y^2 = 4*Area^2/g, X = 2p+d, K = (q^2-d^2)/(4g), J = q^2(q^2-d^2)/(4g) and g = 4 if 16|(q^2-d^2) else g = 1. Other constraints on these primes (see Links) will only permit the following valid pairings:-
prime(n) == 3 mod 4 and a(n) == 1 mod 4
prime(n) == 1 mod 4 and a(n) == 3 mod 4 and prime(n) > a(n)
prime(n) == 1 mod 4 and a(n) == 1 mod 4.
Comments