20200909, 15:59  #1 
May 2017
ITALY
5^{2}×19 Posts 
"New" same approach that isn't factorization
a * b = N
if N mod 4 = 1 then 2 * N + 2 * a ^ 2 + ((ba) / 2) ^ 2 = ((3 * a + b) / 2) ^ 2 now i found that in some cases (i don't know which ones) this is also true 2 * N + 2 * 1 ^ 2 + y ^ 2  ((3 * a + b) / 2) ^ 2 = 0 So for example for the case N = 65 you would 2 * 65 + 2 * a ^ 2 + ((ba) / 2) ^ 2 = ((3 * a + b) / 2) ^ 2 , a * b = 65 , 130 + 2 * 1 ^ 2 + y ^ 2  ((3 * a + b) / 2) ^ 2 = 0 Could you help me: 1) When is this true? 2 * N + 2 * 1 ^ 2 + y ^ 2  ((3 * a + b) / 2) ^ 2 = 0 2) How would you fix the system? 
20200909, 19:59  #2 
Bamboozled!
"πΊππ·π·π"
May 2003
Down not across
10398_{10} Posts 

20200909, 20:13  #3 
May 2017
ITALY
5^{2}×19 Posts 

20200909, 20:17  #4 
"Curtis"
Feb 2005
Riverside, CA
4506_{10} Posts 

20200909, 20:32  #5 
If I May
"Chris Halsall"
Sep 2002
Barbados
3·5^{5} Posts 

20200909, 21:13  #6 
6809 > 6502
"""""""""""""""""""
Aug 2003
101Γ103 Posts
3·19·157 Posts 

20200910, 00:47  #7  
Aug 2006
2^{2}×3^{3}×5×11 Posts 
\(ab \equiv 1 \pmod4,\) so either \(a \equiv b \equiv 1 \pmod4\) or \(a \equiv b \equiv 3 \pmod4\).
Quote:
Yep, this checks out, both as an identity and with the relevant quantities as multiples of 4. Are you asking for which y this equality holds? 

20200910, 07:19  #8 
May 2017
ITALY
5^{2}×19 Posts 
What characteristic must N have for that equality to be true
for example for N = 121 this 2 * 121 + 2 * 1 ^ 2 + y ^ 2 (22) ^ 2 = 0 it's not true that is, y is not integer Last fiddled with by Alberico Lepore on 20200910 at 07:22 
20200910, 07:25  #9 
Undefined
"The unspeakable one"
Jun 2006
My evil lair
3×13×151 Posts 

20200910, 08:32  #10  
May 2017
ITALY
5^{2}·19 Posts 
Quote:
390644893234047643=4*K+3 390644893234047643*3=1171934679702142929=4*H+1 Now I need to understand what k values this system returns integer values a*b=1171934679702142929*(4*k+1) , 2*1171934679702142929*(4*k+1)+2*a^2+((ba)/2)^2z^2=0 , 2*1171934679702142929*(4*k+1)+2*1^2+y^2z^2=0 Last fiddled with by Alberico Lepore on 20200910 at 08:34 

20200910, 08:37  #11 
Undefined
"The unspeakable one"
Jun 2006
My evil lair
3×13×151 Posts 

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