The Modular[RightPseudoQuotient](Ore1, Ore2, p, A, 'm', 'R') calling sequence computes the right pseudo-quotient of Ore1 and Ore2 modulo the prime p. If the fifth (optional) argument is present, it is assigned the multiplier. If the sixth (optional) argument is present, it is assigned the right pseudo-remainder of  Ore1  and Ore2.

The Modular[RightPseudoRemainder](Ore1, Ore2, p, A, 'm', 'Q') calling sequence computes the right pseudo-remainder of Ore1 and Ore2 modulo the prime p. If the fifth (optional) argument is present, it is assigned the multiplier. If the sixth (optional) argument is present, it is assigned the right pseudo-quotient of Ore1  and Ore2.

The Modular[RightQuotient](Ore1, Ore2, p, A, ''R') calling sequence computes the right quotient of Ore1 and Ore2 modulo the prime p. If the fifth (optional) argument is present, it is assigned the right remainder of  Ore1  and Ore2.

The Modular[RightRemainder](Ore1, Ore2, p, A, 'Q') calling sequence computes the right remainder of Ore1 and Ore2 modulo the prime p. If the fifth (optional) argument is present, it is assigned the right quotient of Ore1  and Ore2.

$\mathrm{with}\left(\mathrm{OreTools}\right)\:$

$A≔\mathrm{SetOreRing}\left(n\,\mathrm{differential}\right)$

$A≔\mathrm{UnivariateOreRing}\left(n\,\mathrm{differential}\right)$

$\mathrm{Ore1}≔'\mathrm{OrePoly}'\left(-n\,-\left(-5n+n^2+3\right)\,n-3\,38n^2-1\right)$

$\mathrm{Ore1}≔\mathrm{OrePoly}\left(-n\,-n^2+5n-3\,n-3\,38n^2-1\right)$

$\mathrm{Ore2}≔'\mathrm{OrePoly}'\left(-n\,-\left(-5n+n^2+3\right)\,n-3\right)$

$\mathrm{Ore2}≔\mathrm{OrePoly}\left(-n\,-n^2+5n-3\,n-3\right)$

$\mathrm{Modular}\left[\mathrm{RightPseudoRemainder}\right]\left(\mathrm{Ore1}\,\mathrm{Ore2}\,11\,A\right)$

$\mathrm{OrePoly}\left(5n^5+8n^4+3n^3+n^2+8n+3\,5n^6+5n^5+10n^4+2n^3+6n^2+6n+1\right)$

$Q≔\mathrm{Modular}\left[\mathrm{RightPseudoQuotient}\right]\left(\mathrm{Ore1}\,\mathrm{Ore2}\,19\,A\,'m'\,'R'\right)$

$Q≔\mathrm{OrePoly}\left(18n+7\,18n+3\right)$

$l≔\mathrm{Modular}\left[\mathrm{ScalarMultiply}\right]\left(m\,\mathrm{Ore1}\,19\right)$

$l≔\mathrm{OrePoly}\left(18\left(n^2+13n+9\right)n\,18\left(n^2+13n+9\right)\left(n^2+14n+3\right)\,\left(n^2+13n+9\right)\left(n+16\right)\,18n^2+6n+10\right)$

$r≔\mathrm{Modular}\left[\mathrm{Multiply}\right]\left(Q\,\mathrm{Ore2}\,19\,A\right)$

$r≔\mathrm{OrePoly}\left(n^2+13n+16\,n^3+10n^2+5n+13\,n^3+10n^2+8n+11\,18{\left(n+16\right)}^2\right)$

$\mathrm{Modular}\left[\mathrm{Minus}\right]\left(\mathrm{Modular}\left[\mathrm{Minus}\right]\left(l\,r\,19\right)\,R\,19\right)$

$\mathrm{OrePoly}\left(0\right)$

$R≔\mathrm{Modular}\left[\mathrm{RightRemainder}\right]\left(\mathrm{Ore1}\,\mathrm{Ore2}\,11\,A\,'Q'\right)$

$R≔\mathrm{OrePoly}\left(\frac{5n^4+n^3+6n^2+8n+10}{n+8}\,\frac{5n^5+9n^4+4n^3+3n^2+4n+7}{n+8}\right)$

$r≔\mathrm{Modular}\left[\mathrm{Multiply}\right]\left(Q\,\mathrm{Ore2}\,11\,A\right)$

$r≔\mathrm{OrePoly}\left(\frac{6n^4+10n^3+4n^2+6n+1}{n+8}\,\frac{6n^5+2n^4+6n^3+5n^2+2}{n+8}\,n+8\,5\left(n+3\right)\left(n+8\right)\right)$

$\mathrm{Modular}\left[\mathrm{Minus}\right]\left(\mathrm{Modular}\left[\mathrm{Minus}\right]\left(\mathrm{Ore1}\,r\,11\right)\,R\,11\right)$

$\mathrm{OrePoly}\left(0\right)$

$\mathrm{Modular}\left[\mathrm{RightQuotient}\right]\left(\mathrm{Ore1}\,\mathrm{Ore2}\,19\,A\right)$

$\mathrm{OrePoly}\left(\frac{18n+7}{n^2+13n+9}\,\frac{18}{n+16}\right)$