Abstract
Shnirelman's theorem is applied to solving Diophantine equations, and also discussing of the problems of a representation of Gaussian integers by a sum of odd Gaussian primes.
Introduction
In our note we study the systems of Diophantine equations and connect this problem to Shnirelman's theorem: "Every integer n > 7 is a sum at most s 0 (Shnirelman's constant) odd primes" [1] . The problem of the representation of Gaussian integers by a sum of odd Gaussian primes is explored from an angle of Shnirelman's theorem. Proof. The proof uses induction. Let the theorem be true up to some n−2 = a+b. Let the theorem be false for n = a ′ + b ′ . By Shnirelman's theorem with s 0 = 4, there exist p ≥ q ≥ r ≥ l ∈ π Z 2 such that n = a ′ + b ′ = p + q + r + l. Let us consider 4 cases:
Diophantine Equations and Shnirelman's Theorem
• Case 1: b ′ ≤ l, then we have a solution:
Diophantine Equations and Shnirelman's Theorem
• Case 4: q+r+l < b ′ < p+q+r+l, then we have a solution:
Diophantine Equations and Shnirelman's Theorem
From this, it follows that the theorem is true.
Diophantine Equations and Shnirelman's Theorem
Theorem 2. Given a system of diophantine equations where a > 6 and a ≥ b > 0:
Diophantine Equations and Shnirelman's Theorem
Proof. The proof mimics the one used for theorem 1.
Diophantine Equations and Shnirelman's Theorem
Based on these two theorems, we can propose the following conjecture which will be useful to use when working with Gaussian integers. Conjecture 1. Given the system of equations where a > 6 and a ≥ b > 0:
Diophantine Equations and Shnirelman's Theorem
Let us consider z ∈ Z[i] and let z = a + ib. If the above conjecture is true, then it implies that z is a sum of at most k 1 odd Gaussian primes with non-negative real and imaginary parts [2] .
Gaussian Integers, Odd Gaussian Primes and Shnirelman's Theorem
Let us introduce the notion of "odd" and "even" Gaussian integers, where z is odd if z ≡ 1 (mod 1+i) and even if z ≡ 0 (mod 1 + i).
Our global experiment was to see whenever any element from the set Γ
Some of our computations and results are presented in appendix.
Our global experiment was to see whenever any element from the set Γ
Our experimental work has allowed us to observe some interesting facts about Gaussian integers, notably given z ∈ Γ G , and either Im(z) = Re(z) − 1 or Im(z) = Re(z), z cannot be represented as a sum of odd primes from Γ π . This has led us to consider a different choice for our set
Our global experiment was to see whenever any element from the set Γ
It is worth noting that in both cases Γ G is a maximal set of non-associated Gaussian integers.
Our global experiment was to see whenever any element from the set Γ
Studying the results from [2] and our experimental data we have formulated the following conjecture:
Our global experiment was to see whenever any element from the set Γ
then for all z in A, such that max(Re(z), Im(z)) ≥ 7, z is a sum of at most 3 odd primes from the set K π .
Our global experiment was to see whenever any element from the set Γ
In our study of this conjecture, we have formulated the following proposition:
Our global experiment was to see whenever any element from the set Γ
Proof. The proof is by induction. Let the proposition be true for some k 0 . Let it be false for k 0 + 1, therefore there exists such w that w ≡ k 0 + 1 (mod 1 + i) and max(Re(w), Im(w)) ≥ c 0 + 3 for which the proposition fails. Let Im(w) ≥ c 0 + 3, then w − 3i ≡ k 0 (mod 1 + i) and max(Re(w − 3i), Im(w − 3i)) ≥ c 0 contradicting the inductive hypothesis. The case Re(w) ≥ c 0 + 3 is analogous.
Our global experiment was to see whenever any element from the set Γ
If we assume that every odd Gaussian integer z where Re(z) > 0, Im(z) > 0 and max(Re(z), Im(z)) ≥ c 0 is a sum of 3 odd Gaussian primes with non-negative real and imaginary parts, then we can use proposition 1 to get the following theorem:
Our global experiment was to see whenever any element from the set Γ
If we consider Z as a subset of Z[i], there are even elements of Z which can not be represented as a sum of 2 elements from the set π Z ∩ π Z[i] . However, if one of the hypotheses below is true then the elements of Z as a subset of Z[i] could be represented as a sum of at most s 0 primes from π Z ∩ π Z[i] .
Our global experiment was to see whenever any element from the set Γ
• Hypothesis 1: There exists c 0 > 0 such that every integer n ≥ c 0 , n ≡ 2 (mod 4) is a sum of 2 primes where p i ≡ 3 (mod 4)
Our global experiment was to see whenever any element from the set Γ
• Hypothesis 2: There exists c 0 > 0 such that every integer n ≥ c 0 , n ≡ 1 (mod 4) is a sum of 3 primes where p i ≡ 3 (mod 4)
Our global experiment was to see whenever any element from the set Γ
• Hypothesis 3: There exists c 0 > 0 such that every integer n ≥ c 0 , n ≡ 0 (mod 4) is a sum of 4 primes where p i ≡ 3 (mod 4)
Our global experiment was to see whenever any element from the set Γ
• Hypothesis 4: There exists c 0 > 0 such that every integer n ≥ c 0 , n ≡ 3 (mod 4) is a sum of 5 primes where p i ≡ 3 (mod 4)
Our global experiment was to see whenever any element from the set Γ
Indeed if at least one of the above hypotheses is true, then the following theorem is also true:
Our global experiment was to see whenever any element from the set Γ
Theorem 4. For all n ∈ Z where n ≥ c 1 and c 1 is a constant, there exists m such that m i=1 x i = n has a solution (p 1 , ..., p m ) where ∀i p i ≡ 3 mod 4, and p i is prime and m takes the values ranging from 2 to s 1 (constant).
Our global experiment was to see whenever any element from the set Γ
Proof. Let hypothesis 2 be true and consider the following 3 cases:
Our global experiment was to see whenever any element from the set Γ
• ∀n where n ≡ 0 (mod 4) ∧ n ≥ c 0 + 3 is a sum of four odd primes. Since n − 3 ≡ 1 (mod 4) and n − 3 ≥ c 0 the proof follows hypothesis 2.
Our global experiment was to see whenever any element from the set Γ
• ∀n where n ≡ 3 (mod 4) ∧ n ≥ c 0 + 6 is a sum of five odd primes. This is true due to the previous case.
Our global experiment was to see whenever any element from the set Γ
• ∀n where n ≡ 2 (mod 4) ∧ n ≥ c 0 + 9 is a sum of six odd primes. This is true due to the previous case.
Our global experiment was to see whenever any element from the set Γ
From these cases, we can conclude that c 1 = c 0 + 9 and s 1 = 6.
Our global experiment was to see whenever any element from the set Γ
Using conjecture 2 and theorem 4 we can formulate the following hypothesis:
Our global experiment was to see whenever any element from the set Γ
Conjecture 3. For all z ∈ Γ G , where max(Re(z), Im(z)) ≥ c 0 , z is a sum of at most k 0 odd primes from the set
Our global experiment was to see whenever any element from the set Γ
Based on our experimental evidence, we can also propose the following conjecture:
Our global experiment was to see whenever any element from the set Γ
Conjecture 4. For all z ∈ Γ G , where max(Re(z), Im(z)) ≥ 6, z is a sum of at most 3 odd Gaussian primes from the set S π = {z ∈ π Z[i] | Re(z) ≥ 0 ∧ − Re(z) < Im(z)} (i.e. z = k l=1 p l where p l ∈ S π is an odd Gaussian prime and ||p l || < ||z||).
Discussion and Conclusion
In the present note we have constructed a solution of a system of Diophantine equations of the first degree in several variables over a set of non-negative integers by the Shnirelman theorem. We have demonstrated this approach on Gaussian integers. Based on experimental data on the representation of Gaussian integers by a sum of Gaussian primes we have shown that there is an infinite number of elements of a maximal set of non-associated Gaussian integers Γ G which cannot be represented as a sum of two or three Gaussian primes belonging to a set K π (Γ G ⊂ K π ⊂ Z[i]). In this case we have formulated the Goldbach conjecture for Z[i] as some analogy of the Shnirelman theorem. However, according to our experimental evidence we expect that any Gaussian integer of Γ G , excepting a finite number, can represent by a sum at most three odd Gaussian primes belonging to a set S π (Γ G ⊂ K π ⊂ S π ⊂ Z[i]) where S π is a minimal subset of Z[i] to satisfy such conditions. 
Appendix
p, q, r ∈ Γ π ; max(||p||, ||q||, ||r||) < ||z|| Representation z = 9 p = 3 q = 3 r = 3 z = p + q + r z = 13 p = 7 q = 3 r = 3 z = p + q + r z = 17 p = 11 q = 3 r = 3 z = p + q + r z = 21 p = 11 q = 7 r = 3 z = p + q + r z = 25 p = 11 q = 11 r = 3 z = p + q + r z = 29 p = 19 q = 7 r = 3 z = p + q + r z = 33 p = 19 q = 7 r = 7 z = p + q + r z = 37 p = 19 q = 11 r = 7 z = p + q + r z = 7 p = 3 + 2i q = 2 − i r = 2 − i z = p + q + r z = 11 p = 7 q = 2 + i r = 2 − i z = p + q + r z = 15 p = 8 + 3i q = 5 − 2i r = 2 − i z = p + q + r z = 19 p = 11 q = 4 + i r = 4 − i z = p + q + r z = 23 p = 13 + 2i q = 7 r = 3 − 2i z = p + q + r z = 27 p = 13 + 2i q = 11 r = 3 − 2i z = p + q + r z = 31 p = 23 q = 4 + i r = 4 − i z = p + q + r z = 35 p = 23 q = 6 + i r = 6 − i z = p + q + r z = 6 + i p = 2 + i q = 2 + i r = 2 − i z = p + q + r z = 6 + 3i p = 2 + i q = 2 + i r = 2 + i z = p + q + r z = 7 + 2i p = 3 q = 2 + i r = 2 + i z = p + q + r z = 7 + 4i p = 3 + 2i q = 2 + i r = 2 + i z = p + q + r z = 8 + i p = 2 + i q = 3 r = 3 z = p + q + r z = 8 + 3i p = 3 + 2i q = 3 r = 2 + i z = p + q + r z = 8 + 5i p = 3 + 2i q = 3 + 2i r = 2 + i z = p + q + r z = 9 + 2i p = 3 + 2i q = 3 r = 3 z = p + q + r z = 9 + 4i p = 5 + 2i q = 2 + i r = 2 + i z = p + q + r z = 9 + 6i p = 5 + 4i q = 2 + i r = 2 + i z = p + q + r z = 19 + 16i p = 10 + 9i q = 6 + 5i r = 3 + 2i z = p + q + r z = 27 + 24i p = 23 + 22i q = 2 + i r = 2 + i z = p + q + r z = 39 + 36i p = 35 + 34i q = 2 + i r = 2 + i z = p + q + r z = 48 + 45i p = 25 + 24i q = 20 + 19i r = 3 + 2i z = p + q + r z = 50 + i p = 23 q = 20 + i r = 7 z = p + q + r z = 50 + 3i p = 40 + i q = 6 + i r = 4 + i z = p + q + r z = 50 + 7i p = 40 + i q = 5 + 4i r = 5 + 2i z = p + q + r z = 50 + 11i p = 36 + 5i q = 9 + 4i r = 5 + 2i z = p + q + r z = 50 + 15i p = 35 + 8i q = 10 + 3i r = 5 + 4i z = p + q + r z = 50 + 19i p = 35 + 16i q = 10 + i r = 5 + 2i z = p + q + r z = 50 + 25i p = 35 + 18i q = 10 + 3i r = 5 + 4i z = p + q + r z = 7 + 3i

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