This repo provides sum of four squares for Mersenne primes.
All Mersenne primes besides M₁ do not have a sum of three squares representation.
Currently sum of four squares are provided for M₂..M₂₉ in M.py.
validate.gp and validate.py demonstrate how to access M.py from Pari/GP and Python.

Validation

pi@raspberrypi5:~/Mersenne_primes_sum4squares $ make

time -f %E\\n  gp -q < validate.gp
all asserts OK
0:00.02

time -f %E\\n  python validate.py
all asserts OK
0:00.05

pi@raspberrypi5:~/Mersenne_primes_sum4squares $ 

Table

M_n=2^p_n-1=a²+b²+c²+d², #x short for #digits(x)

Sum of 4 squares was computed with Pari/GP script.
The script takes d to be maximal odd number less than sqrt(M_n).
Nothing can be said about numbers a, b and c other than a²+b²+c²=M_n-d².

With N=M_n-d² script determined minimal k, such that k*4*N+2*N-1 is prime.
This allows to create a ternary quadratic form by the method from Dirichlet;
which allows to compute a,b and c with N=a²+b²+c² easily.

Single thread runtime of Pari/GP script on AMD 7950X CPU is reported as t.

Runtimes of current script definitely do not allow to complete table up to M₅₁.

n	#Mn		#a	#b	#c	#d		k	t[s]	alg
1	1
2	1		1	1	1	1		2	1
3	2		1	1	1	1		2	1
4	3		1	1	1	2		2	1
5	4		2	2	1	2		1	1
6	6		2	2	1	3		2	1
7	6		2	1	1	3		6	1
8	10		3	3	3	5		5	1
9	19		5	4	4	10		6	1
10	27		7	7	7	14		10	1
11	33		8	8	8	17		16	1
12	39		10	10	9	20		4	1
13	157		40	39	39	79		139	1
14	183		46	46	46	92		4	1
15	386		96	97	96	193		93	1
16	664		167	166	166	332		216	1
17	687		172	172	171	344		2	1
18	969		241	242	242	485		155	1
19	1,281		321	320	320	641		553	2
20	1,332		334	333	333	666		45	1
21	2,917		730	729	730	1,459		475	9
22	2,993		748	749	748	1,497		1,330	27
23	3,376		844	844	844	1,688		618	18
24	6,002		1,501	1,501	1,501	3,001		741	82
25	6,533		1,634	1,634	1,634	3,267		3,853	540
26	6,987		1,747	1,747	1,747	3,494		7,570	12616
27	13,395		3,348	3,349	3,349	6,698		6,845	5377
28	25,962		6,491	6,490	6,490	12,981		5,759	23278
29	33,265		8,317	8,316	8,316	16,633		10,667	82982	(200)
30	39,751		19,875	19,875	19,875	19,876		(31G)	315	foursquares.gp
31	65,050		32,525	32,525	32,525	32,525		(31G)	861	"
32	227,832		113,916	113,916	113,916	113,915		(31G)	14198	"
33	258,716
34	378,632
...
50	23,249,425
51	24,862,048
n	#Mn		#a	#b	#c	#d		k	t[s]	alg

foursquares.gp

foursquares.gp