$2p-2$ as the sum of consecutive prime numbers

Question

Progress:

Let $p$ be a prime such that $p≡1$ (mod 6) then $2p-2$ can be written uniquely (up to the order of addends) as the sum of some consecutive prime numbers.

These are first ten examples:

$$2⋅7-2=12=5+7$$ $$2⋅13-2=24=11+13$$ $$2⋅19-2=36=17+19$$ $$2⋅31-2=60=29+31$$ $$2⋅37-2=72=5+7+11+13+17+19$$ $$2⋅43-2=84=41+43$$ $$2⋅61-2=120=59+61$$ $$2⋅67-2=132=13+17+19+23+29+31$$ $$2⋅73-2=144=71+73$$ $$2⋅79-2=156=17+19+23+29+31+37$$

I don't even have the slightest idea that how and from where to start the proof. Hope you guys can help me. Thanks in advance.

Answer 1

Let's try the next prime that is $1 \pmod{6}$, i.e. $p = 97$.

We want either $2$ or $6$ consecutive primes which sum to $2 \cdot 97 - 2 = 192$.

Since $89,97,101$ are consecutive primes and $89+97 = 186 < 192 < 198 = 97+101$, we see that we cannot write $192$ as the sum of $2$ consecutive primes.

Since $19,23,29,31,37,41,43$ are consecutive primes and $19+23+29+31+37+41 = 180 < 192 < 204 = 23+29+31+37+41+43$, we see that we cannot write $192$ as the sum of $6$ consecutive primes.

Therefore, $p = 97$ is a counterexample to your conjecture.

Answer 2

Try $p = 157, 223, 337, 397, 439, \ldots$.

Answer 3

There are many counterexamples to your conjecture. The numbers below cannot be written as the sum of consecutive prime numbers:

$ 157, 223, 337, 397, 439, 487, 499, 631, 673, 709, 739, 751, 757, 769, 787, 907, 967, 997, 1069, 1087, 1117, 1171, 1237, 1249, 1297, 1399, 1423, 1447, 1471, 1543, 1567, 1579, 1657, 1777, 1831, 1861, 1867, 1987, 2011, 2017, 2137, 2179, 2203, 2221, 2251, 2287, 2347, 2371, 2377, 2389, 2437, 2467, 2503, 2617, 2677, 2683, 2707, 2749, 2767, 2833, 2917, 2953, 3019, 3049, 3061, 3067, 3163, 3187, 3217, 3229, 3319, 3433, 3499, 3517, 3571, 3631, 3643, 3691, 3697, 3727, 3739, 3847, 3877, 3889, 3907, 3943, 4027, 4201, 4297, 4363, 4441, 4507, 4561, 4567, 4591, 4597, 4621, 4657, 4813, 4861, 4909, 4951, 4987, 4999, 5059, 5077, 5167, 5179, 5227, 5323, 5431, 5449, 5527, 5557, 5569, 5581, 5623, 5737, 5749, 5779, 5791, 5821, 5839, 5857, 5953, 6037, 6067, 6079, 6121, 6211, 6217, 6247, 6373, 6379, 6397, 6427, 6469, 6481, 6547, 6577, 6607, 6619, 6637, 6673, 6679, 6967, 6997, 7027, 7039, 7057, 7177, 7207, 7297, 7321, 7369, 7411, 7417, 7477, 7537, 7603, 7639, 7669, 7699, 7753, 7867, 7933, 7963, 7993, 8017, 8101, 8167, 8179, 8191, 8209, 8287, 8311, 8317, 8329, 8353, 8419, 8443, 8467, 8563, 8581, 8689, 8707, 8713, 8719, 8737, 8779, 8887, 8893, 8923, 8929, 9103, 9109, 9127, 9133, 9157, 9181, 9187, 9199, 9277, 9391, 9397, 9403, 9511, 9547, 9601, 9613, 9619, 9643, 9649, 9661, 9697, 9733, 9739, 9781, 9787, 9817, 9829, 9871, 9901, 9907, 9973, 10099, 10111, 10159, 10177, 10243, 10321, 10357, 10399, 10453, 10477, 10513, 10567, 10597, 10651, 10663, 10687, 10729, 10753, 10771, 10789, 10837, 10867, 10957, 10993, 11047, \ldots$