Solution

Let us allow the value \(x=0\) as well; we prove the same statement under this more general constraint. Obviously that implies the statement with the original conditions.

Call a pair \(\{p, q\}\) of primes with \(p \neq q\) special if \(p q=x^{2}+x+k\) for some nonnegative integer \(x\). The following claim is the key mechanism of the problem:

Claim:

a. For every prime \(r\), there are at most two primes less than \(r\) forming a special pair with \(r\).
b. If such \(p\) and \(q\) exist, then \(\{p, q\}\) is itself special.

We present two proofs of the claim.

Proof 1: We are interested in integers \(1 \leqslant x<r\) satsfying

\[\begin{equation*} x^{2}+x+k \equiv 0 \quad(\bmod r) \tag{1} \end{equation*}\]

Since there are at most two residues modulo \(r\) that can satisfy that quadratic congruence, there are at most two possible values of \(x\). That proves (a). Now suppose there are primes \(p, q\) with \(p<q<r\) and nonnegative integers \(x, y\) such that

\[\begin{aligned} & x^{2}+x+k=p r \\ & y^{2}+y+k=q r \end{aligned}\]

From \(p<q<r\) we can see that \(0 \leqslant x<y \leqslant r-1\). The numbers \(x, y\) are the two solutions of (1); by Vieta's formulas, we should have \(x+y \equiv-1(\bmod r)\), so \(x+y=r-1\). Letting \(K=4 k-1, X=2 x+1\), and \(Y=2 y+1\), we obtain

\[\begin{gathered} 4 p r=X^{2}+K \\ 4 q r=Y^{2}+K \end{gathered}\]

with \(X+Y=2 r\). Multiplying the two above equations,

\[\begin{aligned} 16 p q r^{2} & =\left(X^{2}+K\right)\left(Y^{2}+K\right) \\ & =(X Y-K)^{2}+K(X+Y)^{2} \\ & =(X Y-K)^{2}+4 K r^{2} \\ 4 p q & =\left(\frac{X Y-K}{2 r}\right)^{2}+K \end{aligned}\]

In particular, the number \(Z=\frac{X Y-K}{2 r}\) should be an integer, and so \(4 p q=Z^{2}+K\). By parity, \(Z\) is odd, and thus

\[p q=z^{2}+z+k \quad \text { where } z=\frac{Z-1}{2}\]

so \({p, q}\) is special. *Proof 2:* As before, we suppose that

\[\begin{aligned} & x^{2}+x+k=p r \\ & y^{2}+y+k=q r \end{aligned}\]

Subtracting, we have

\[(x+y+1)(x-y)=r(p-q)\]

As before, we have \(x+y=r-1\), so \(x-y=p-q\), and

\[\begin{aligned} & x=\frac{1}{2}(r+p-q-1) \\ & y=\frac{1}{2}(r+q-p-1) \end{aligned}\]

Then,

\[\begin{aligned} k=p r-x^{2}-x & =\frac{1}{4}\left(4 p r-(r+p-q-1)^{2}-2(r+p-q-1)\right) \\ & =\frac{1}{4}\left(4 p r-(r+p-q)^{2}+1\right) \\ & =\frac{1}{4}\left(2 p q+2 p r+2 q r-p^{2}-q^{2}-r^{2}+1\right) \end{aligned}\]

which is symmetric in \(p, q, r\), so

\[pq = z^{2}+z+k \quad \text { where } z=\frac{1}{2}(p+q-r-1)\]

and \({p, q}\) is special. Now we settle the problem by induction on \(|S|\), with \(|S| \leqslant 3\) clear. Suppose we have proven it for \(|S|=n\) and consider \(|S|=n+1\). Let \(r\) be the largest prime in \(S\); the claim tells us that in any valid cycle of primes: - the neighbors of \(r\) are uniquely determined, and - removing \(r\) from the cycle results in a smaller valid cycle. It follows that there is at most one valid cycle, completing the inductive step. *Comment:* The statement is not as inapplicable as it might seem. For example, for \(k=41\), the following 385 primes form a valid cycle of primes:

53, 4357, 104173, 65921, 36383, 99527, 193789, 2089123, 1010357, 2465263, 319169, 15559, 3449, 2647, 1951, 152297, 542189, 119773, 91151, 66431, 222137, 1336799, 469069, 45613, 1047941, 656291, 355867, 146669, 874879, 2213327, 305119, 3336209, 1623467, 520963, 794201, 1124833, 28697, 15683, 42557, 6571, 39607, 1238833, 835421, 2653681, 5494387, 9357539, 511223, 1515317, 8868173, 114079681, 59334071, 22324807, 3051889, 5120939, 7722467, 266239, 693809 , 3931783, 1322317, 100469, 13913, 74419, 23977, 1361, 62983, 935021, 512657, 1394849, 216259, 45827, 31393, 100787, 1193989, 600979, 209543, 357661, 545141, 19681, 10691, 28867, 165089, 2118023, 6271891, 12626693, 21182429, 1100467, 413089, 772867, 1244423, 1827757, 55889, 1558873, 5110711, 1024427, 601759, 290869, 91757, 951109, 452033, 136471, 190031, 4423, 9239, 15809, 24133, 115811, 275911, 34211, 877, 6653, 88001, 46261, 317741, 121523, 232439, 379009, 17827, 2699, 15937, 497729, 335539, 205223, 106781, 1394413, 4140947, 8346383, 43984757, 14010721, 21133961, 729451, 4997297, 1908223, 278051, 529747, 40213, 768107, 456821, 1325351, 225961, 1501921, 562763, 75527, 5519, 9337, 14153, 499, 1399, 2753, 14401, 94583, 245107, 35171, 397093, 195907, 2505623, 34680911, 18542791, 7415917, 144797293, 455529251, 86675291, 252704911, 43385123, 109207907, 204884269, 330414209, 14926789, 1300289, 486769, 2723989, 907757, 1458871, 65063, 4561, 124427, 81343, 252887, 2980139, 1496779, 3779057, 519193, 47381, 135283, 268267, 446333, 669481, 22541, 54167, 99439, 158357, 6823, 32497, 1390709, 998029, 670343, 5180017, 13936673, 2123491, 4391941, 407651, 209953, 77249, 867653, 427117, 141079, 9539, 227, 1439, 18679, 9749, 25453, 3697, 42139, 122327, 712303, 244261, 20873, 52051, 589997, 4310569, 1711069, 291563, 3731527, 11045429, 129098443, 64620427, 162661963, 22233269, 37295047, 1936969, 5033449, 725537, 1353973, 6964457, 2176871, 97231, 7001, 11351, 55673, 16747, 169003, 1218571, 479957, 2779783, 949609, 4975787, 1577959, 2365007, 3310753, 79349, 23189, 107209, 688907, 252583, 30677, 523, 941, 25981, 205103, 85087, 1011233, 509659, 178259, 950479, 6262847, 2333693, 305497, 3199319, 9148267, 1527563, 466801, 17033, 9967, 323003, 4724099, 14278309, 2576557, 1075021, 6462593, 2266021, 63922471, 209814503, 42117791, 131659867, 270892249, 24845153, 12104557, 3896003, 219491, 135913, 406397, 72269, 191689, 2197697, 1091273, 2727311, 368227, 1911661, 601883, 892657, 28559, 4783, 60497, 31259, 80909, 457697, 153733, 11587, 1481, 26161, 15193, 7187, 2143, 21517, 10079, 207643, 1604381, 657661, 126227, 372313, 2176331, 748337, 64969, 844867, 2507291, 29317943, 14677801, 36952793, 69332267, 111816223, 5052241, 8479717, 441263, 3020431, 1152751, 13179611, 38280013, 6536771, 16319657, 91442699, 30501409, 49082027, 72061511, 2199433, 167597, 317963, 23869, 2927, 3833, 17327, 110879, 285517, 40543, 4861, 21683, 50527, 565319, 277829, 687917, 3846023, 25542677, 174261149, 66370753, 9565711, 1280791, 91393, 6011, 7283, 31859, 8677, 10193, 43987, 11831, 13591, 127843, 358229, 58067, 15473, 65839, 17477, 74099, 19603, 82847, 21851, 61.