| Displaying 1-3 of 3 results found.
page
1
a(n) gives one fourth of the even leg of the second of the two Pythagorean triangles with hypotenuse A080109(n) = A002144(n)^2. The odd leg is given in A253804.
+20
3
5, 30, 34, 145, 111, 180, 371, 330, 876, 1560, 1746, 505, 1635, 840, 3014, 3570, 5181, 2249, 1710, 7980, 1379, 3435, 10920, 7230, 2056, 8970, 14490, 11240, 4981, 3900
COMMENTS
See A253804 for comments and the Dickson reference.
REFERENCES
L. E. Dickson, History of the Theory of Numbers, Carnegie Institution, Publ. No. 256, Vol. II, Washington D.C., 1920, p. 227.
EXAMPLE
n = 7: A080175(7) = 7890481 = 53^4 = 2809^2; A002144(7)^4 = A253804(7)^2 + (4*a(7))^2 = 2385^2 + (4*371)^2. The other Pythagorean triangle with hypotenuse 53^2 = 2809 has odd leg A253802(7) = 1241 and even leg 4* A253303(7) = 4*630 = 2520: 53^4 = 1241^2 + (4*630)^2.
a(n) gives the odd leg of one of the two Pythagorean triangles with hypotenuse A080109(n) = A002144(n)^2. This is the smaller of the two possible odd legs.
+10
3
7, 65, 161, 41, 1081, 369, 1241, 671, 721, 3471, 959, 9401, 4681, 1695, 3281, 7599, 10199, 24521, 3439, 18335, 37241, 45241, 24465, 29281, 64001, 18561, 31855, 27761, 76601, 7825
COMMENTS
The corresponding even legs are given in 4* A253803. The legs of the other Pythagorean triangle with hypotenuse A080109(n) are given A253804(n) (odd) and A253805(n) (even). Each fourth power of a prime of the form 1 (mod 4) (see A002144(n)^2 = A080175(n)) has exactly two representations as sum of two positive squares (Fermat). See the Dickson reference, (B) on p. 227. This means that there are exactly two Pythagorean triangles (modulo leg exchange) for each hypotenuse A080109(n) = A002144(n)^2, n >= 1. See the Dickson reference, (A) on p. 227. Note that the Pythagorean triangles are not always primitive. E.g., n = 2: (65, 4*39, 13^2) = 13*(5, 4*3, 13). For each prime congruent 1 (mod 4) ( A002144) there is one and only one such non-primitive triangle with hypotenuse p^2 (just scale the unique primitive triangle with hypotenuse p with the factor p). Therefore, one of the two existing Pythagorean triangles with hypotenuse from A080109 is primitive and the other is imprimitive.
REFERENCES
L. E. Dickson, History of the Theory of Numbers, Carnegie Institution, Publ. No. 256, Vol. II, Washington D.C., 1920, p. 227.
EXAMPLE
n = 7: A080175(7) = 7890481 = 53^4 = 2809^2; A002144(7)^4 = a(7)^2 + (4* A253803(7))^2 = 1241^2 + (4*630)^2. The other Pythagorean triangle with hypotenuse 53^2 = 2809 has odd leg A253804(7) = 2385 and even leg 4* A253305(7) = 4*371 = 1484: 53^4 = 2385^2 + (4*371)^2.
a(n) gives one fourth of the even leg of one of the two Pythagorean triangles with hypotenuse A080109(n) = A002144(n)^2. The odd leg is given in A253802(n).
+10
3
6, 39, 60, 210, 210, 410, 630, 915, 1320, 1780, 2340, 990, 2730, 3164, 4620, 5215, 5610, 4290, 8145, 8106, 2730, 6630, 12116, 12540, 4080, 17485, 17451, 18480, 9690, 24414
COMMENTS
See A253802 for comments and the Dickson reference.
REFERENCES
L. E. Dickson, History of the Theory of Numbers, Carnegie Institution, Publ. No. 256, Vol. II, Washington D.C., 1920, p. 227.
EXAMPLE
n = 7: A080175(7) = 7890481 = 53^4 = 2809^2; A002144(7)^4 = A253802(7)^2 + (4*a(7))^2 = 1241^2 + (4*630)^2. The other Pythagorean triangle with hypotenuse
53^2 = 2809 has odd leg A253804(7) = 2385 and even leg 4* A253305(7) = 4*371 = 1484: 53^4 = 2385^2 + (4*371)^2.
Search completed in 0.009 seconds
| 