**From the computation of Erdös-Woods Numbers to the quadratic Goldbach Conjecture**

N. Lygeros

Erdös-Woods Numbers are defined as the length of an interval of consecutive integers whose every element is not coprime with its extremeties. Woods was the first to find such numbers, Dowe proved there exists an infinity and Cégielski, Heroult and Richard that their set is recursive. Our aim is to study the arithmetical proprieties of those numbers in relation with Goldbach Conjecture.
For x less than 100000 k=3 is sufficient. For primes less than 23 there are four ambiguous cases for k=2 : (x+1,x+2)=(2,3) or (8,9) ; (6,7) or (48,49) ; (14,15) or (224,225); (75,76) or (1215,1216). The first three of these are members of the infinite family : (2^n-2,2^n-1),(2^n(2^n-2),(2^n-1)^2)
2184,2185,2186,2187,2188,2189,2190,2191,2192,2193,2194,2195,2196,2197,2198,2199,2200
Idea : Primality of consecutive numbers.
Idea : Primality of consecutive numbers. Remark : d can be prime.
Idea : For every prime p, p^n + 1 belongs to this set.
Idea : Bertrand-Tchebychev's Theorem
(i) for any integer i, 0 (ii) for 0 Idea : Analogous to the Cantor Theorem on consecutive primes in arithmetic progression.
16,22,34,36,46,56,64,66,70,76,78,86,88,92,94,96, 100,106,112,116,118,120,124,130,134,142,144,146, 154,160,162,186,190,196,204,210,216,218,220,222, 232,238,246,248,250,256,260,262,268,276,280,286, 288,292,296,298,300,302,306,310,316,320,324,326, 328,330,336,340,342,346,356,366,372,378,382,394, 396,400,404,406,408,414,416,424,426,428,430,438, 446,454,456,466,470,472,474,476,484,486,490,494, 498,512,516,518,520,526,528,532,534,536,538,540.
Let EW(n) be the n-th Erdös-Woods number. EW(1) = 16, EW(2) = 22, EW(3) = 34,...
* d_0=1 * For all 0 * For all 0 * It exists i
Ideas 903, [[41, 39], [431, 22], [59, 43], [211, 84], [173, 139], [73, 58], [83, 27]] 2545, [[53, 13], [89, 4], [307, 135], [373, 207], [181, 45], [197, 67], [587, 100]] 4533, [[103, 48], [443, 331], [409, 252], [1031, 92]]
* d_0=1 * d_1=d+1 * For all 1 * For all 1 * It exists i Remark 1: Vsemirnov's conjecture is false for 26^2 and 34^2 (by direct computation) Remark 2: As 8^2, 14^2 and 20^2 are Erdös-Woods numbers the theorem can be only sufficient.
16, [[5, 1], [11, 6]] 36, [[7, 1], [29, 6]] 100, [[11, 4], [89, 30]] 144, [[13, 4], [131, 75]] 256, [[17, 1], [239, 221]] 324, [[19, 10], [61, 31], [263, 60]] 484, [[23, 4], [461, 206]] 784, [[29, 14], [151, 40], [211, 16], [191, 41], [593, 15]] 900, [[31, 3], [79, 52], [821, 364]] 1296, [[37, 17], [1259, 989]] 1600, [[41, 17], [1559, 1055]] 1764, [[43, 11], [1721, 403]] 2116, [[47, 4], [2069, 1068]]
In other words, if a contrained quadratic goldbach condition holds then d^2 is an Erdös-Woods number. This CQGC implies the last condition of the theorem. This also means that the quadratic goldbach condition would not be sufficient to prove the theorem because we need a special goldbach partition. On the other side, this theorem could lead to a new strategy for the proof of a special case of Goldbach conjecture with the quadratic formulation.
n=2: 34,36 n=3: 92,94,96 n=4: 216,218,220,222 n=5: 532,534,536,538,540 n=6: 1834,1836,1838,1840,1842,1844 n=7: 2166,2168,2170,2172,2174,2176,2178 n=8: 4312,4314,4316,4318,4320,4322,4324,4326 |