Concordant Form
المؤلف:
Dickson, L. E
المصدر:
History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Dover
الجزء والصفحة:
...
18-5-2020
949
Concordant Form
A concordant form is an integer triple 
where
![<span style=]() {a^2+b^2=c^2; a^2+Nb^2=d^2, " src="https://mathworld.wolfram.com/images/equations/ConcordantForm/NumberedEquation1.gif" style="height:44px; width:99px" /> |
(1)
|
with 
and 
integers. Examples include
![<span style=]() {14663^2+111384^2=112345^2; 14663^2+47·111384^2=763751^2
{1141^2+13260^2=13309^2; 1141^2+53·13260^2=96541^2
{2873161^2+2401080^2=3744361^2; 2873161^2+83·2401080^2=22062761^2. " src="https://mathworld.wolfram.com/images/equations/ConcordantForm/NumberedEquation2.gif" style="height:150px; width:253px" /> |
(2)
|
Dickson (2005) states that C. H. Brooks and S. Watson found in The Ladies' and Gentlemen's Diary (1857) that 
and 
can be simultaneously squares for 
only for 1, 7, 10, 11, 17, 20, 22, 23, 24, 27, 30, 31, 34, 41, 42, 45, 49, 50, 52, 57, 58, 59, 60, 61, 68, 71, 72, 74, 76, 77, 79, 82, 85, 86, 90, 92, 93, 94, 97, 99, and 100 (which evidently omits 47, 53, and 83 from above). The list of concordant primes less than 1000 is now complete with the possible exception of the 16 primes 103, 131, 191, 223, 271, 311, 431, 439, 443, 593, 607, 641, 743, 821, 929, and 971.
REFERENCES:
Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Dover, p. 475, 2005.
MathPages. "Concordant Forms." https://www.mathpages.com/home/kmath286.htm.
الاكثر قراءة في نظرية الاعداد
اخر الاخبار
اخبار العتبة العباسية المقدسة
