7
$\begingroup$

This question is related to A conjecture about an unlimited path and Any odd number is of form $a+b$ where $a^2+b^2$ is prime but I present it on its own if anyone would like to help finding counterexamples.

Conjecture:

For all odd numbers $n>159$ there are positive integers $a\le b$ such that:

  1. $\quad n=a+b$
  2. $\quad a^2+b^2$ and $a^2+(b+2)^2$ are primes

Let $S_n=\{a\in\mathbb Z^+|\exists b\in\mathbb Z^+:n=a+b\wedge a^2+b^2\in\mathbb P\}$. Below there is a list of $n$ and $S_n\cap S_{n+2}$, the set of the possible $a\,$ that fulfills the conjecture for the odd number $n$, $1<n<1000$ (0 denotes the empty set):

3 {1}
5 {1,2}
7 {2}
9 0 
11 {3,5}
13 {4}
15 {1,2,7}
17 {2,7}
19 0 
21 {5}
23 {9}
25 {1,7}
27 {5,10}
29 {5,10}
31 {5}
33 {4,13}
35 {2,6,12}
37 {2,5,10}
39 {5,10}
41 {3,8,13}
43 {8,13}
45 {16,22}
47 {2,15,22}
49 0 
51 {8,20,25}
53 {4,8,18,23}
55 {1,17}
57 {7,17}
59 {15}
61 {23}
63 {8,19,23,29}
65 {11,12,16,17,21}
67 {2,20}
69 {10,20}
71 {13,20}
73 {13,14,29,34}
75 0 
77 {5,17,25,27,30}
79 {5,25}
81 {8,25}
83 {4,9,14,18,23,24,39}
85 {7,26,32,41}
87 {2,32}
89 {40}
91 {40,43}
93 {4,28,29,34}
95 {11,17,26,37,41}
97 {2,17}
99 {5}
101 {3,5,18,35,38,50}
103 {19,43,44}
105 {26}
107 {17,25,37,40,45,50}
109 {10,35,50}
111 {5,10,20,23,28,35,50}
113 {3,18,28,39,48,54}
115 {47}
117 {2,10,40,47,55}
119 {10,30,40}
121 {13,23,50,58}
123 {4,13,19,38,58}
125 {1,21,27,47}
127 {35,47,50,55}
129 {5,20}
131 {15,45,53,60,65}
133 {43,53}
135 {31,32,37,56}
137 {2,5,10,22,32,42,52,57}
139 {5}
141 {5,25,28,50,58}
143 {3,23,24,34,53,54,64}
145 {11,17,37,41,62}
147 {10,17,20,32,37,50,67}
149 {30}
151 {5,40,53,65}
153 {8,74}
155 {12,17,26,46,52,57,66,71,72}
157 {40,52,67,77}
159 0 
161 {10,13,43,50,65,78}
163 {23,28,58,68,74}
165 {31,32,61,76}
167 {12,32,37,50}
169 {5}
171 {10,23,43,65,68,85}
173 {19,23,29,43,48,59,68,73}
175 {16,17,22,82}
177 {7,40,55}
179 {35,40,50}
181 {38,65,83}
183 {8,34,38,49,53,83}
185 {6,21,36,52,56,62,76,82,91}
187 {20,37,40,50,62}
189 {20,50,55,85}
191 {3,18,23,35,68,70}
193 {14,19,23,34,53,58,68,79,94}
195 {46,47,92}
197 {20,27,40,55,60,72,75,92,95}
199 {20,95}
201 {20,25,43,65,73,85}
203 {3,38,53,54,64,93,94}
205 {1,52,86,91}
207 {5,32,82}
209 {5,40,45,75,80}
211 {13,25,53,70,103}
213 {13,44,59,88}
215 {11,37,46,61,72,76,81,102,106}
217 {2,25,55,65,107}
219 {5,10,25,40}
221 {3,20,23,25,33,40,50,63,70,83,98}
223 {4,23,88,98,104}
225 {26,41,46,52,67,86,91,97,107}
227 {7,17,52,55,65,72,77,95,97}
229 {85}
231 {68,83,85,103}
233 {4,14,39,48,59,74}
235 {2,26,31,62,116}
237 {65,70,97}
239 {35,65,70,95}
241 {13,80,88,100,113}
243 {13,58,64,104}
245 {2,11,16,47,51,71,101,122}
247 {7,10,40,47,115,122}
249 {50,115}
251 {8,65,78,85,98,100,123}
253 {59,74,98,113}
255 {2,7,26,52,112}
257 {12,25,52,60,72,85,100,102,125}
259 {85}
261 {23,28,40,103,130}
263 {23,24,34,38,49,63,69,99,109,118}
265 {32,37,46,56,62,71,82,92}
267 {2,20,22,32,55,95,100,107,127}
269 {25,45,50,55,75,100}
271 {8,25,55,68,73,88,100,103,115,128}
273 {8,23,73,83,103,128}
275 {36,42,47,92,111,112}
277 {2,10,20,65,67,97,112,125,127}
279 {20,25,55}
281 {18,25,28,38,55,68,73,78,83,113,118,135}
283 {29,64,68,73,79,89,118}
285 {22,86,116}
287 {5,45,47,57,62,67,127,130}
289 {40,130}
291 {40,83,98,115,118,128}
293 {4,13,19,24,29,39,98,109,128}
295 {61,67,106,136,142}
297 {10,32,47,62,67,112,145}
299 {5,120,145}
301 {23,65,95,125,130,143,145}
303 {4,34,58,89,104,113,119}
305 {17,21,32,46,71,81,82,116,117,132,136}
307 {25,40,47,107,122}
309 {20,25,40,100,125}
311 {8,20,30,40,70,90,98,110,113,125,138}
313 {23,38,44,88,104,118}
315 {37,71,76,101,142}
317 {5,10,17,30,97,102,135,137,152}
319 {5,10,50,85}
321 {5,50,58,70,83,115,128,140,155}
323 {23,44,49,58,63,74,88,118,129,138,148,159}
325 {37,76,92,106,122,151}
327 {20,40,50,67,82,92,115,137}
329 {55,75,125,130,145}
331 {28,38,55,58,85,113,125,130,133,145}
333 {14,29,38,83,113,139}
335 {11,12,22,26,31,37,42,66,77,102,111,116,126,142,156,161}
337 {7,32,77,85,92,100,152,155,160}
339 {5,70,145}
341 {5,10,23,60,73,83}
343 {19,58,83,124,134,139,143}
345 {61,67,97,101,121,146,152,172}
347 {5,25,42,45,75,87,90,102,120,132,172}
349 {10,40,110}
351 {80,88,118,133,148,158}
353 {8,13,59,99,119,133,158,164}
355 {26,67,76,127,137,146,166,172}
357 {20,25,55,67,137,172}
359 {20,55,60,85,100,105,170}
361 {5,53,85}
363 {14,53}
365 {27,52,86,91,106,132,151,166}
367 {10,17,37,40,92,112,125,157,170,175}
369 {20,85,110}
371 {5,15,43,45,55,65,83,88,93,113,145,150,160,165}
373 {13,49,68,83,94,133,139,169,179}
375 {2,31,32,37,47,61,76,86,107,142,151,166,187}
377 {7,10,20,25,27,30,60,90,92,102,142,155,160,185}
379 {80,95,130,160}
381 {73,95,130,140,143,160,178}
383 {3,8,73,74,78,113,123,124,169}
385 {1,122,146,151}
387 {5,22,47,70,112,155}
389 {5,25,65,70}
391 {13,43,53,70,103,130,140,148}
393 {4,13,43,49,53,58,64,103,134,148,188}
395 {37,56,71,87,97,126,131,137,166,167}
397 {97,160,197}
399 {80}
401 {5,35,38,58,63,68,73,80,105,123,145,153,175,188,193}
403 {14,29,38,43,44,73,109,128,164,179,193}
405 {17,41,62,82,131,151,172}
407 {15,17,35,42,65,80,97,107,117,152,162,175}
409 {35,65,95,110}
411 {8,13,20,40,53,65,85,113,145,200}
413 {3,8,13,24,39,53,64,109,124,163,164,183,194,198,204}
415 {47,61,137,142,161}
417 {10,47,52,67,110,140,142,170,187}
419 {95,105,110,140,180}
421 {5,35,85,100,110,115,118,128,140,163}
423 {13,43,44,59,74,83,128,133,163,179}
425 {2,11,16,32,52,106,117,127,156,162,191,207}
427 {2,25,32,50,67,92,115,145,152,185}
429 {5,50,70,115,160}
431 {28,43,63,68,78,95,118,140,143,160,183,193,208}
433 {34,68,89,94,134,169,179,184,188,193,208}
435 {22,32,52,107,136,151,191}
437 {20,25,70,85,90,117,155,157,165,195}
439 {215}
441 {38,118,220}
443 {13,18,19,78,88,173,179,184}
445 {17,26,31,71,91,92,97,112,116,121,131,142,182,212}
447 {5,7,92,95,115,127,130,200,212,215,217}
449 {5,20,30,85,160,175,200,210,215}
451 {23,25,35,58,100,103,178,193,200}
453 {38,44,73,79,103,149,178}
455 {2,11,36,51,62,81,116,146,157,186,192}
457 {7,10,62,107,130,137,157,160,202,205,215}
459 {35,80,130,160,190}
461 {35,43,55,68,80,93,100,110,120,138,168,178,188,190,208,210,225,228}
463 {73,74,109,118,139,143,169,208}
465 {1,11,46,47,52,112,116,131,142,157,196,206,221}
467 {12,60,65,97,117,137,157,160,177}
469 {40,145,160}
471 {8,38,53,70,95,128,148,190,235}
473 {48,49,59,63,104,113,124,148,174,208}
475 {17,31,32,46,67,122,136,137,167,206,221}
477 {2,22,65,155,172,185,187,205,217,230}
479 {90,100,165,210,230,235}
481 {5,10,50,73,158,193,223,233}
483 {8,13,34,68,79,158,163,164,223}
485 {11,32,41,72,86,107,117,126,136,146,152,171,186,227,231,237}
487 {25,35,47,82,110,137,185}
489 {25,145,230}
491 {8,25,48,60,70,73,90,100,120,130,135,138,163,180,198,230}
493 {8,19,59,109,194,199}
495 {92,101,106,136,157,236}
497 {12,30,50,75,102,122,135,157,160,162,190,205,207,222}
499 {85,115,205,230}
501 {28,133,163,185,193,205,215}
503 {23,58,74,103,109,139,154,168,174,184,193,213,218,233}
505 {31,76,122,131,146,187,191,197,211}
507 {5,37,50,70,77,107,122,160,170,185,212,215,232,250}
509 {15,145,150,250}
511 {85,100,125,163,178,205,230}
513 {13,163,178,194,199,214,224,233}
515 {16,41,61,81,107,116,131,142,152,161,182,217,226,241}
517 {50,107,142,152,155,172,232}
519 {5,40,190,215}
521 {13,65,93,105,118,135,148,158,190,218,225,238,255,260}
523 {13,89,103,118,164,218,239}
525 {26,61,82,101,127,166,226,227,251}
527 {10,32,37,72,75,80,97,122,140,145,152,162,165,197,210,212,227,240,247}
529 {35,80,145,220}
531 {35,38,50,58,158,188,190,200,220,223,230,233,250}
533 {24,29,63,94,98,109,119,163,184,213,218,223,249,254,259,264}
535 {17,47,91,211,232,242,247}
537 {10,52,80,130,160,167,185,197,215,227,232,247}
539 {20,90,130,145,155,160,215}
541 {5,8,20,28,100,130,133,145,148,155,158,170,235}
543 {8,13,14,29,133,139}
545 {6,12,17,31,52,56,86,87,112,117,137,162,182,216,241,256,267}
547 {47,62,65,95,142,170,187,197,247,260,265}
549 {25,50,65,70,115,125,130,160,170}
551 {33,45,78,88,113,115,123,135,183,205,225,240}
553 {23,68,89,139,173,193,208,244}
555 {67,101,136,166,187,197,272}
557 {5,45,62,70,87,90,100,107,115,152,157,167,220,235,270,272}
559 {25,50,70,115,185,205}
561 {53,115,118,125,173,185,190,193,203,205,215}
563 {29,34,43,58,59,63,69,74,93,98,104,153,154,173,193,204,219,228,238,258,263,268}
565 {22,52,121,137,256,272,277,281}
567 {67,130,142,152,235}
569 {50,100,245,255,265}
571 {38,95,125,160,173,233,253,260,265}
573 {8,14,28,109,148,173,179,208,229,233,238,244}
575 {17,32,157,182,186,196,206,212,232,257,271}
577 {17,22,32,35,115,137,157,175,187,212,247,257,260,280}
579 {100,260}
581 {10,48,90,93,130,148,153,178,180,193,200,233,243,260}
583 {8,23,28,113,118,148,149,158,193,214,224,268,269}
585 {11,67,122,157,166,172,181,197,212,226,241,242}
587 {10,17,20,50,55,85,130,140,177,195,212,222,232,242,245,275}
589 {10,20,35,250,275}
591 {35,38,73,100,103,110,145,158,223,263}
593 {73,74,94,124,163,169,174,183,194,209,234,243,249,254,263}
595 {47,86,131,142,151,157,172,232,271}
597 {47,92,122,157,247,280,290}
599 {30,65,205,240,285}
601 {10,40,103,125,128,143,148,155,188,208,233,245,283}
603 {43,49,74,79,113,169,188,199,284}
605 {7,26,57,62,82,96,117,192,201,227,241,267,276}
607 {2,10,62,167,172,197,227,230,275,292,295}
609 {25,110,170,220}
611 {15,25,30,33,38,68,70,73,80,83,88,133,138,145,153,170,173,218,223,298}
613 {34,59,88,128,133,218,223,224,248,263}
615 {2,11,56,71,107,167,176,182,202,232,241,266,292,296}
617 {12,17,45,85,95,142,147,165,172,267,282,285,292,305}
619 {35,70,130,215,265}
621 {8,13,25,130,220,293,295}
623 {8,13,34,59,64,69,99,123,129,148,183,188,199,243,253,293,299}
625 {26,106,127,181,226,236,251,257,272,287}
627 {47,50,82,125,127,155,167,200,212,235,245,260}
629 {10,20,50,115,165,245,270,290}
631 {23,35,50,58,110,115,128,145,178,193,238,250,275,278}
633 {103,104,128,143,154,179,194,218,224,278}
635 {1,16,81,92,152,157,167,191,197,232,261,282,291}
637 {80,115,167,185,215,242,262,265,317}
639 {10,170,185,245,250,275}
641 {8,20,55,73,78,100,128,138,158,165,185,225,228,240,258,275,280,318}
643 {38,73,158,173,194,203,233,289}
645 {1,16,67,107,122,151,187,196,202,236,281,287}
647 {25,27,42,87,125,195,207,230,235,265,312}
649 {125,190,200,235}
651 {73,95,110,118,173,200,218,220,235,275}
653 {13,18,23,44,73,103,138,144,173,174,203,229,244,248,279,294}
655 {7,11,52,71,76,136,137,166,176,187,227,256,271}
657 {52,67,82,107,130,137,175,197,205,280,310,322}
659 {55,130,145,155,175,195,270,285,310,315}
661 {28,53,55,70,118,125,133,140,145,155,298,310}
663 {53,64,79,118,164,179,268,278,298,313}
665 {2,17,26,41,47,66,106,117,121,131,157,166,177,212,216,222,241,257,262,267,271,297,326}
667 {2,17,52,82,172,205,217,257,262,277,292}
669 {125,160,215,230,320}
671 {48,60,118,120,135,185,215,233,235,248,313}
673 {29,34,128,133,143,193,328}
675 {17,22,46,56,77,82,142,151,157,182,332}
677 {17,62,67,75,117,127,130,155,157,247,260,262,272,300,305,317,325,335}
679 {65,260}
681 {20,65,98,110,163,170,230,253,308}
683 {3,9,19,24,39,43,54,83,139,148,163,213,214,219,224,264,303,308,329}
685 {31,37,41,61,71,76,107,127,136,142,161,172,187,197,226,232,257,266,271,272,337}
687 {112,127,187,200,242,272,307,322}
689 {5,15,20,55,75,90,105,200,225,255,290,340}
691 {20,85,208,218,268,290,293,310,323,340}
693 {4,13,59,128,149,158,188,218,229,283,299,323,334}
695 {6,42,47,61,81,117,121,132,162,171,191,251,262,276,302,331}
697 {20,40,80,100,130,145,182,302,307,325}
699 {40,310,325}
701 {40,108,125,135,160,170,173,183,188,203,240,255,273,293,325}
703 {4,14,28,53,109,169,184,203,223,239,283}
705 {26,97,151,166,167,181,187,251,316,317,346}
707 {10,32,52,65,100,160,180,205,222,270,300}
709 {10,55,85,100,160,185,295,335}
711 {50,58,88,110,113,185,233}
713 {19,48,53,73,103,158,168,224,233,239,243,284,304,309,314,353,354}
715 {1,112,161,212,277,281,302,307,311}
717 {7,40,202,212,232,245,250,262,277,302,320,332,355}
719 {115,180,250,295,320}
721 {8,25,85,160,188,220,295,305,358}
723 {13,34,79,124,158,163,169,208,259,268,328,334,338,343,358}
725 {16,22,107,122,131,137,156,161,207,237,246,252,267,282,286,287,302,341,351}
727 {5,22,82,212,220,230,235,265,280,325}
729 {125,140,175,220,235,290,325}
731 {18,40,53,63,125,140,148,173,193,200,203,208,220,240,245,248,270,273,298,318,338,350,355}
733 {19,83,88,89,134,169,173,193,254,274,319,338,349}
735 {17,52,71,101,122,221,251,296}
737 {12,35,65,80,115,135,140,142,197,295,305,315,350}
739 {35,40,145,230,305,320}
741 {20,73,83,145,158,160,163,215,218,248,265,283,290,293,313,358}
743 {4,28,39,49,79,99,114,119,138,163,193,219,263,264,284,289,313,323,329,364}
745 {2,11,32,37,107,121,152,167,182,236,247,257,286,331,347}
747 {20,52,110,152,242,275,277,305,340,370}
749 {25,80,125,170,195,230,265,340,345,360,365}
751 {13,25,70,73,115,125,130,133,143,163,170,233,278,328,338,365}
753 {13,73,74,143,149,173,214,233,254,274,278,343,359,364}
755 {2,52,57,106,132,147,181,187,242,307,316,322,326,327,356,361}
757 {10,52,62,112,200,265,355}
759 {50,100,140,265,355,365}
761 {13,45,65,85,100,120,145,153,158,163,178,180,198,230,260,265,270,283,338,340,345,355}
763 {38,58,158,184,188,214,278,283,298,304,353}
765 {47,71,92,101,112,161,172,211,256,257,266,296,307}
767 {17,47,72,75,85,105,115,142,190,220,285,307,310,352,355,365,375,382}
769 {55,85,115,145,190,235,260,310}
771 {85,115,145,193,218,223,250,323,325,340}
773 {4,9,38,48,74,94,108,149,163,233,259,299,333}
775 {22,52,107,137,142,146,181,187,191,256,272,281,332,337,386}
777 {65,107,122,187,215,230,260,317,337}
779 {25,100,135,170,230,255,280,305,345}
781 {5,8,73,83,188,200,223,235,268,305,325,383}
783 {8,19,83,149,188,209,214,229,253,278,329,383}
785 {32,37,56,61,66,96,97,117,121,166,167,182,212,226,241,261,266,277,306,312,336,351,352,371}
787 {2,70,85,95,110,167,187,212,250,262,292,380}
789 {55,380}
791 {8,15,18,45,68,73,100,115,220,243,253,285,288,295,300,348,353,360,373}
793 {14,98,133,154,164,199,209,218,229,233,254,259,289,304,314,353,379,388}
795 {16,47,52,56,86,101,157,196,206,232,307,316,332,341,361,391}
797 {30,42,82,87,117,145,147,150,175,205,212,240,270,310,312,322,362,367}
799 {25,145,215,295,350}
801 {8,25,70,85,100,148,185,203,223,263,280,295,298,340}
803 {3,39,74,148,204,258,269,298,328,369,398}
805 {137,191,242,251,286,397}
807 {10,47,65,67,125,130,182,220,262,265,335,392,395}
809 {10,125,130,190,300,320,330,375,395}
811 {10,43,113,128,130,178,208,238,325,383,395,400}
813 {29,43,68,113,119,128,323,373,379}
815 {32,97,101,106,116,276,286,311,331,336,357,377,406}
817 {32,97,142,170,320,352,365,367}
819 {25,125,145,170,185,215,320,340}
821 {95,98,125,150,170,203,250,263,268,288,300,323,340,345,368,388,390,410}
823 {4,8,34,188,203,223,229,263,268,299,323,334,364,368,373}
825 {17,71,86,91,142,151,197,211,241,377,406}
827 {22,80,85,120,122,152,157,165,187,197,207,215,217,220,237,255,275,287,317,320,377}
829 {130,170,215,310}
831 {58,130,143,155,163,215,248,265,268,295,313,365}
833 {38,54,58,123,148,173,174,183,234,248,279,298,353,359,369,389,394,398,409}
835 {52,76,212,221,271,272,326,376,382,407}
837 {5,52,82,160,175,182,187,200,230,332,347,407}
839 {10,35,55,70,115,160,165,200,225,295,405}
841 {10,35,38,68,103,133,193,253,295,323,383,388}
843 {14,29,73,103,133,148,179,224,233,244,259,284}
845 {27,37,51,72,76,92,107,116,172,212,226,241,246,277,317,327,332,337,362,367,376}
847 {2,20,50,127,247,310,320,332,355,410}
849 {35,50,80,260,290,310,340,410}
851 {13,25,30,35,48,98,103,110,120,168,175,190,240,260,275,290,308,350,363,410,418,420}
853 {28,34,53,74,119,139,164,248,268,274,308,313,403}
855 {37,61,112,116,146,167,176,182,232,326,367,371,386,401,427}
857 {47,85,87,105,110,112,122,135,160,177,305,342,345,360,370,382,395}
859 {25,110,170,260,265,290,355,370,395,415}
861 {25,38,83,170,200,208,218,235,248,335,355,395,403,428}
863 {4,43,59,83,149,154,178,213,219,248,254,293,334,349,374,384,429}
865 {22,41,62,91,121,172,226,232,356,392,397,412,431}
867 {10,25,32,52,67,70,97,142,155,185,197,217,232,295,347,370,377,380,412}
869 {80,105,120,185,200,380,425}
871 {23,73,80,128,133,160,233,245,275,290,293,298,313,380,395,400,428}
873 {128,164,184,244,289,293,349,359,368,418}
875 {97,101,146,151,166,167,177,191,197,216,256,267,282,327,332,386,397,407}
877 {22,32,65,95,97,112,145,227,230,275,277,332,347,355,385}
879 {55,130,145,230,265,275,280,350,385,395,430}
881 {33,53,83,118,130,168,178,213,230,263,280,298,310,313,393,395,398,403,415,430,440}
883 {4,14,103,104,169,203,254,259,298,403,428,434}
885 {31,56,91,122,166,281,292,382,416}
887 {45,62,65,82,102,110,115,117,122,155,160,192,235,247,285,365,387,430}
889 {115,335,365}
891 {70,115,130,175,205,260,340,343,358,368}
893 {18,23,44,93,103,123,143,158,168,174,198,214,219,224,243,244,249,263,273,343,348,373,414,419}
895 {31,37,71,86,106,116,131,146,166,236,317,361,386,392,401,406,421}
897 {25,77,137,172,220,257,280,337,365,407,410}
899 {10,15,55,125,150,185,195,215,260,300,320,365}
901 {10,88,110,125,185,188,193,205,325,338,358,383,400,433}
903 {19,59,104,143,178,253,254,319,338,359,389}
905 {2,7,26,47,57,67,116,157,171,176,177,222,241,247,276,291,311,326,392,412,432,441,442,447,451}
907 {112,157,167,187,247,272,295,320,332,347,350,395}
909 {160,320,370}
911 {18,58,73,75,90,130,140,148,158,160,190,205,238,255,278,318,335,338,355,378}
913 {23,148,233,283,338,413,454}
915 {11,26,92,97,121,127,206,232,277,326,331,382,412,422}
917 {5,32,82,97,202,225,247,277,302,377,382,395,410,425,445}
919 {80,155,250,295,340,370,400,425,445}
921 {68,113,133,155,223,233,248,250,283,295,328,340,398,425,440}
923 {14,44,108,124,159,164,183,199,218,223,224,229,258,283,309,328,339,369,393,454}
925 {86,91,92,127,137,232,271,326,431,452}
927 {55,127,152,160,232,242,290,295,305,440,452}
929 {25,30,55,65,165,215,345,370,445}
931 {53,55,83,158,220,265,293,298,320}
933 {53,79,113,164,193,203,259,268,269,293,299,304,358,359,389,434,464}
935 {26,27,91,111,112,152,161,202,212,217,307,316,332,347,372,381,397,441,467}
937 {17,77,142,145,170,182,200,212,242,322,325,355,455}
939 {5,40,125,140,170,220,290,325,395,455,460}
941 {13,153,163,240,305,375,383,388,395,420,455,468}
943 {13,109,113,143,148,169,188,323,353,388,418}
945 {32,41,62,136,232,286,296,317,367,401,412}
947 {32,42,62,90,105,175,190,197,240,242,255,282,297,327,357,372,385,402,405,445}
949 {80,85,340,385,415,445}
951 {20,43,65,85,88,140,185,250,290,325,328,353,358,403,440,443,475}
953 {29,43,49,88,99,133,134,149,159,188,199,229,234,268,273,284,294,308,309,318,323,328,333,353,379,384,418,464}
955 {17,32,47,56,82,107,136,146,202,211,241,247,281,397,431,442,467}
957 {5,40,92,107,122,125,265,340,397,425,430,442}
959 {125,265,365,445}
961 {13,35,85,98,103,118,173,253,260,278,290,365,368,433,448}
963 {13,49,64,73,118,119,134,169,173,208,268,283,368,374,403,424,433,439}
965 {21,31,86,102,136,186,222,247,271,317,357,372,401,447,452,461}
967 {10,40,200,215,230,317,362,382,400,422,452}
969 {55,145,200,215,230,295,370,400}
971 {18,23,55,80,88,110,123,143,193,200,223,230,233,253,295,335,338,360,370,373,400,408,425,438,460,465}
973 {58,74,103,148,149,223,239,298,379,383,388,404,424,428,479}
975 {62,131,176,187,196,206,271,281,301,316,322,332,346,427,472}
977 {7,10,17,30,62,95,112,120,160,210,222,237,270,310,322,332,387,390,417,432,437,442,472}
979 {125,160,205,260,310,355,400}
981 {25,28,58,68,110,133,155,178,208,275,298,328,355,368,380,433}
983 {28,29,73,178,208,228,229,233,243,254,304,308,313,328,378,389,398,399,424,429,444,449}
985 {22,26,31,151,172,202,352,377,412,421}
987 {20,40,52,95,142,265,352,377,397}
989 {10,65,125,160,255,265,270,340,365,415,450,485}
991 {10,65,110,115,143,173,278,323,368,370,428,473,478}
993 {19,104,109,124,143,173,299,308,404,443,449,478}
995 {6,7,61,76,101,107,117,127,146,152,192,217,221,241,256,266,296,327,341,382,387,406,416,442,446,466,472,496}
997 {97,107,127,130,152,217,232,310,337,367,395,422,425,442,460}
999 {230,395,400,430,470}

So far tested for $n<50,000$.

$\endgroup$
  • 1
    $\begingroup$ the difference between $a^2+b^2$ and $a^2+ (b+2)^2$ is $4(b+1)$ maybe that will help narrow down b. $\endgroup$ – user451844 Aug 13 '17 at 12:40
  • $\begingroup$ @RoddyMacPhee: Yes, this conjecture predicts two primes with the gap $4(b+1)$. $\endgroup$ – Lehs Aug 13 '17 at 12:54
  • $\begingroup$ maybe you can prove that all a and b pairs that sum to n allow $a^2+b^2$ and $a^2+(b+2)^2$ to cover all possible prime placements in a range that is big enough such that at lest one of those placement must be prime? $\endgroup$ – user451844 Aug 13 '17 at 12:56
  • $\begingroup$ and n that has no overlap would produce about n numbers that are 1 mod 4 if these are in sequence does it go high enough to force one pair of them to be prime ? $\endgroup$ – user451844 Aug 13 '17 at 13:19
  • 1
    $\begingroup$ $a=5$ and $b=4$ is a solution for n=9. $\endgroup$ – pietfermat Aug 16 '17 at 6:01

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