NSolve and Solve are unable to Solve System of Polynomials with methods available to them

Is this number to big to work with

Solve
[{x^2 + \
((17969491597941066732916128449573246156367561808012600070888918835531\
7264603414909334933722478686507552308558641999292218144366847228740520\
6525793749569434838926317115252252565441098081917061174250970244071801\
0364831638288518852689 + q^2)/
q) x == -\
1796949159794106673291612844957324615636756180801260007088891883553172\
6460341490933493372247868650755230855864199929221814436684722874052065\
2579374956943483892631711525225256544109808191706117425097024407180103\
64831638288518852689,

x^2 + ((p^3 +
1796949159794106673291612844957324615636756180801260007088891\
8835531726460341490933493372247868650755230855864199929221814436684722\
8740520652579374956943483892631711525225256544109808191706117425097024\
40718010364831638288518852689 p)/
p^2) x == \
-179694915979410667329161284495732461563675618080126000708889188355317\
2646034149093349337224786865075523085586419992922181443668472287405206\
5257937495694348389263171152522525654410980819170611742509702440718010\
364831638288518852689 ,

x^2 + (p + q) x +
179694915979410667329161284495732461563675618080126000708889188355\
3172646034149093349337224786865075523085586419992922181443668472287405\
2065257937495694348389263171152522525654410980819170611742509702440718\
010364831638288518852689 == 0, Mod[p + q, 2] == 0}, {x, p, q}]

• There is 4 equations with 3 unknowns... – mmeent May 3 '18 at 7:07
• Anyway, evaluating this in 11.1.1 produces no errors. It just tells me there are no solutions. – mmeent May 3 '18 at 7:08
• @mmeent is it bad to have 4 equation with 3 unknowns do I need another one, also I'm using 11.2 sorry for not letting putting it in – user546733 May 3 '18 at 7:26

You get imaginary solutions.

eqs1 = {x^2 + \
((17969491597941066732916128449573246156367561808012600070888918835531\
7264603414909334933722478686507552308558641999292218144366847228740520\
6525793749569434838926317115252252565441098081917061174250970244071801\
0364831638288518852689 + q^2)/
q) x == -\
1796949159794106673291612844957324615636756180801260007088891883553172\
6460341490933493372247868650755230855864199929221814436684722874052065\
2579374956943483892631711525225256544109808191706117425097024407180103\
64831638288518852689,
x^2 + ((p^3 +
179694915979410667329161284495732461563675618080126000708889\
1883553172646034149093349337224786865075523085586419992922181443668472\
2874052065257937495694348389263171152522525654410980819170611742509702\
440718010364831638288518852689 p)/
p^2) x == \
-179694915979410667329161284495732461563675618080126000708889188355317\
2646034149093349337224786865075523085586419992922181443668472287405206\
5257937495694348389263171152522525654410980819170611742509702440718010\
364831638288518852689,
x^2 + (p + q) x +
17969491597941066732916128449573246156367561808012600070888918835\
5317264603414909334933722478686507552308558641999292218144366847228740\
5206525793749569434838926317115252252565441098081917061174250970244071\
8010364831638288518852689 == 0};


Eliminate x and solve for p

sol1 = First@Solve[Eliminate[eqs1, x], p]

(*   {p ->1796949159794106673291612844957324615636756180801260007088891883\
5531726460341490933493372247868650755230855864199929221814436684722874\
0520652579374956943483892631711525225256544109808191706117425097024407\
18010364831638288518852689/q}   *)

sol2 = First@
Solve[((p /. sol1) + q)/(k*2) == 0 && k \[Element] Integers, q]

(*   {q -> ConditionalExpression[-I \
\[Sqrt]179694915979410667329161284495732461563675618080126000708889188\
3553172646034149093349337224786865075523085586419992922181443668472287\
4052065257937495694348389263171152522525654410980819170611742509702440\
718010364831638288518852689, k \[Element] Integers && k != 0]}   *)

sol3 = Solve[eqs1 /. sol1 /.
Simplify[sol2, k \[Element] Integers && k != 0], x]

(*   {{x -> -I \
\[Sqrt]179694915979410667329161284495732461563675618080126000708889188\
3553172646034149093349337224786865075523085586419992922181443668472287\
4052065257937495694348389263171152522525654410980819170611742509702440\
718010364831638288518852689}, {x ->
I \[Sqrt]\
1796949159794106673291612844957324615636756180801260007088891883553172\
6460341490933493372247868650755230855864199929221814436684722874052065\
2579374956943483892631711525225256544109808191706117425097024407180103\
64831638288518852689}}   *)


to get p, do

p /. sol1 /. sol2


Test

eqs1 /. sol1 /. sol2 /. sol3

(*   {{True, True, True}, {True, True, True}}   *)


Edit

to get all politive and negative solutions for x,p,q you better do

s3 = Solve[
Flatten[{eqs1, {p + q == 0 || (p + q)/(2*k) == 0,
k \[Element] Integers, k != 0}}], {x, p, q}]


It shows, they are independent of k.

• If i know there are suppose to be real solution, is there a way to readjust the equation to find that, and how come in your calculation you took out the Mod[p+q,2]==0 – user546733 May 3 '18 at 11:04