Primes of the forms $4m+1$ and $4m+3$ I ask if it is possible to demonstrate that any prime number $p$ of the form $4m+1$ can be written as $$p=q_1+2q_2=(4n_1+3)+2(4n_2+3)$$ and, conversely, that any prime number $q$ of the form $4n+3$ can be written as $$q=p_1+2p_2=(4m_1+1)+2(4m_2+1)$$
For the first primes we have:
$$3=1+2\cdot1\qquad5=-1+2\cdot3$$
$$7=5+2\cdot1\qquad13=7+2\cdot3$$
$$11=1+2\cdot5\qquad17=3+2\cdot7$$
$$19=17+2\cdot1\qquad29=7+2\cdot11$$
$$23=13+2\cdot5\qquad37=23+2\cdot7$$
Thanks a lot for your attention.
It follows that :
$$p=(4n_1+3)+2(4n_2+3) \iff \color{#d05}{p = 4(n_1+2n_2+2)+1}$$

And

$$q=(4m_1+1)+2(4m_2+1)\iff \color{#2d0}{q = 4(m_1+2m_2)+3}$$

so , $p = 4k+1$ and $q = 4l+3$ regardless of whether $p,q$ are primes or composite.
It follows that :
$$p=(4n_1+3)+2(4n_2+3) \iff \color{#d05}{p = 4(n_1+2n_2+2)+1}$$

And

$$q=(4m_1+1)+2(4m_2+1)\iff \color{#2d0}{q = 4(m_1+2m_2)+3}$$

so , $p = 4k+1$ and $q = 4l+3$ regardless of whether $p,q$ are primes or composite.