A131020 For all cyclic quadrilaterals with four consecutive primes as sides that have an area that is prime after rounding, the sequence gives the first of these four consecutive primes.

3, 5, 13, 17, 61, 67, 97, 139, 157, 163, 173, 223, 271, 349, 353, 419, 479, 503, 541, 691, 701, 743, 877, 941, 1013, 1049, 1051, 1097, 1123, 1229, 1231, 1249, 1297, 1301, 1423, 1453, 1493, 1531, 1559, 1607, 1621, 1697, 1811, 1901, 1999, 2017, 2027, 2053, 2087

Offset: 1

Author

Jonathan Vos Post, Jun 09 2007

Keywords

• easy
• nonn

Comments

The semiperimeters of cyclic quadrilaterals with four consecutive odd prime sides are given in A131019. This arises in the cyclic quadrilateral analog of A106171.

Examples

			a(5) = 61 because (61 + 67 + 71 + 73)/2 = 136 and sqrt((136 - 61)*(136 - 67)*(136 - 71)*(136 - 73)) = 4603.43622 and round(4603.43622) = 4603 is prime.
		

References

• Coxeter, H. S. M. and Greitzer, S. L. "Cyclic Quadrangles; Brahmagupta's Formula", Sect. 3.2 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 56-60, 1967.

Links

• J. L. Coolidge, A Historically Interesting Formula for the Area of a Quadrilateral, Amer. Math. Monthly 46, 345-347, 1939.
• Eric Weisstein's World of Mathematics, Brahmagupta's Formula.

Crossrefs

Cf. A000040, A106171, A131019.

Programs

Maple

Digits := 80 : isA131020 := proc(p) local p2,p3,p4,s,area; if isprime(p) then p2 := nextprime(p) ; p3 := nextprime(p2) ; p4 := nextprime(p3) ; s := (p+p2+p3+p4)/2 ; area := round(sqrt((s-p)*(s-p2)*(s-p3)*(s-p4))) ; RETURN(isprime(area)) ; else false ; fi ; end: for n from 1 to 380 do if isA131020(ithprime(n)) then printf("%d,",ithprime(n)) ; fi ; od;

Formula

a(n) = prime(k) for some k such that, where S = semiperimeter = (prime(k) + prime(k+1) + prime(k+2) + prime(k+3))/2 is an element of A131019 and rounded area = round(sqrt((S-prime(k))*(S-prime(k+1))*(S-prime(k+2))*(S-prime(k+3)))) is prime.

Extensions

Edited by R. J. Mathar, Jun 12 2007