SUMS OF PRODUCTS OF CONGRUENCE CLASSES AND OF ARITHMETIC PROGRESSIONS
2009 ◽
Vol 05
(04)
◽
pp. 625-634
Keyword(s):
Consider the congruence class Rm(a) = {a + im : i ∈ Z} and the infinite arithmetic progression Pm(a) = {a + im : i ∈ N0}. For positive integers a,b,c,d,m the sum of products set Rm(a)Rm(b) + Rm(c)Rm(d) consists of all integers of the form (a+im) · (b+jm)+(c+km)(d+ℓm) for some i,j,k,ℓ ∈ Z. It is proved that if gcd (a,b,c,d,m) = 1, then Rm(a)Rm(b) + Rm(c)Rm(d) is equal to the congruence class Rm(ab+cd), and that the sum of products set Pm(a)Pm(b)+Pm(c)Pm eventually coincides with the infinite arithmetic progression Pm(ab+cd).
2014 ◽
Vol 57
(3)
◽
pp. 551-561
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1974 ◽
Vol 18
(2)
◽
pp. 188-193
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2008 ◽
Vol 51
(1)
◽
pp. 47-56
◽
Keyword(s):
1999 ◽
Vol 42
(1)
◽
pp. 25-36
◽
2011 ◽
Vol 54
(2)
◽
pp. 431-441
◽
2004 ◽
Vol 47
(2)
◽
pp. 191-205
◽
Keyword(s):