¿Existen números primos distintos$p,q,r$ tales que$\frac{p+q+r}{3}; \frac{p+q}2; \frac{r+q}2; \frac{p+r}2$ son números primos.

Question

¿Existen números primos distintos$p,q,r$ tales que$$\frac{p+q+r}{3}; \frac{p+q}2; \frac{r+q}2; \frac{p+r}2$ $ son números primos.

Mi trabajo:

Encontré $(p,q,r)=(5,17,29)$.

¿Hay algún otro ejemplo?

Answer 1

Hay secuencias aritméticas arbitrariamente largas de primos. REFERENCIA
Deje $$ a_0

El mismo principio "Ramsey" nos permite encontrar muchos otros patrones finitos en el conjunto de primos.

Answer 2

Si tiene Mathematica, puede obtener fácilmente cientos de ejemplos si no miles:

{{5,17,29},{5,17,89},{5,17,101},{5,17,197},{5,17,449},{5,17,521},{5,29,53},{5,29,89},{5,29, 269},{5,29,509},{5,41,137},{5,41,173},{5,41,257},{5,41,293},{5,53,461},{5,89,173},{5,89, 449},{5,101,113},{5,101,197},{5,113,353},{5,137,197},{5,137,449},{5,173,293},{5,173, 521},{5,197,269},{5,197,389},{5,197,521},{5,257,461},{5,257,509},{5,269,449},{5,293, 509},{5,353,449},{5,389,449},{7,19,67},{7,31,271},{7,67,127},{7,67,139},{7,67,379},{7, 79,127},{7,79,307},{7,127,439},{7,139,307},{7,211,271},{7,307,439},{11,23,503},{11,47, 71},{11,47,251},{11,71,131},{11,71,491},{11,83,443},{11,83,503},{11,107,191},{11,131, 167},{11,131,251},{11,131,347},{11,131,491},{11,263,443},{11,347,491},{13,61,193},{13,73, 181},{13,73,241},{13,109,349},{13,193,373},{13,241,433},{13,349,409},{17,41,101},{17,41, 461},{17,41,521},{17,101,461},{17,197,257},{17,197,509},{17,257,449},{19,43,439},{19,67, 127},{19,103,199},{19,103,379},{19,139,523},{19,199,283},{19,199,463},{19,283,379},{23, 59,239},{23,71,443},{23,83,311},{23,83,431},{23,191,503},{23,251,443},{29,53,509},{29,89, 173},{29,113,149},{29,113,269},{29,113,449},{29,149,233},{29,197,317},{29,269,509},{29, 353,449},{31,43,103},{31,43,163},{31,43,523},{31,103,283},{31,127,331},{31,163,223},{37, 97,157},{37,241,421},{41,53,173},{41,317,521},{43,79,379},{43,103,523},{43,151,523},{43, 163,283},{43,271,283},{43,379,499},{47,59,467},{47,71,131},{47,131,491},{47,167,179},{47, 167,419},{47,167,467},{47,179,347},{47,227,479},{47,311,491},{47,347,419},{47,419, 467},{53,89,449},{53,113,281},{53,173,293},{53,173,461},{53,449,509},{59,419,443},{59, 479,503},{61,241,421},{61,397,421},{67,79,487},{67,127,487},{67,379,487},{71,131, 431},{71,191,311},{71,227,491},{71,311,467},{71,431,491},{73,241,373},{73,349,409},{79, 139,283},{79,199,223},{79,307,367},{79,379,463},{83,131,383},{83,131,503},{83,191, 263},{83,191,443},{83,251,263},{83,251,383},{83,263,443},{83,263,503},{89,113,269},{89, 113,389},{89,269,449},{101,113,233},{101,197,281},{101,233,353},{101,257,521},{103,151, 463},{103,223,523},{103,283,463},{103,379,439},{107,167,227},{107,167,359},{107,227, 239},{107,227,479},{107,347,359},{109,277,337},{109,277,457},{109,313,349},{113,233, 353},{113,269,449},{113,389,449},{113,389,509},{131,227,491},{137,197,257},{137,197, 389},{137,257,449},{137,317,389},{139,307,487},{149,233,389},{151,211,271},{163,379, 499},{167,191,311},{167,227,359},{167,347,419},{173,281,353},{173,389,449},{193,433, 541},{197,257,269},{227,239,467},{227,251,311},{239,383,479},{241,421,457},{257,281, 521},{269,449,509},{277,337,397},{283,331,463},{317,461,521},{337,409,457},{359,419, 479},{389,449,509},{401,461,521},{419,479,503},{431,467,491}}

Voy a verificar al azar uno de estos:$p = 47, q = 419, r = 467$. Entonces los promedios son:$311, 233, 257, 443$.

Aquí está el código que utilicé: Select[Subsets[Prime[Range[100]], {3}], PrimeQ[(#[[1]] + #[[2]] + #[[3]])/3] && PrimeQ[(#[[1]] + #[[2]])/2] && PrimeQ[(#[[1]] + #[[3]])/2] && PrimeQ[(#[[2]] + #[[3]])/2] &] Si quieres más, simplemente cambia el 100 a 1000 o lo que sea.

Answer 3

Sí. Algunos ejemplos:

• $(p,q,r)=(5, 17, 89)$;
• $(p,q,r)=(7, 19, 67)$;
• $(p,q,r)=(5, 29, 53)$;
• $(p,q,r)=(11, 47, 71)$.