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Primed and Perfect (Posted on 2014-01-03) Difficulty: 3 of 5
Determine the minimum value of a positive integer n > 1 such that n + 6 is a prime number and 9*n + 7 is a perfect square.

What are the next two smallest values of n with this property?

No Solution Yet Submitted by K Sengupta    
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Solution computer solution Comment 1 of 1

10   for N=1 to 999999
20     P=N+6
30     if prmdiv(P)=P then
40          :Sq=9*N+7
50          :Sr=int(sqrt(Sq)+0.5)
60          :if Sr*Sr=Sq then print N,P,Sq,Sr
70             :Ct=Ct+1:if Ct>45 then stop
80   next

finds the first three values of n higher than 1 are 53, 277 and 373:

  n             n + 6           9*n + 7      sqrt(9*n+7)
1               7               16              4
53              59              484             22
277             283             2500            50
373             379             3364            58
641             647             5776            76
821             827             7396            86
1393            1399            12544           112
3061            3067            27556           166
3761            3767            33856           184
4993            4999            44944           212
6293            6299            56644           238
7861            7867            70756           266
9473            9479            85264           292
10133           10139           91204           302
11377           11383           102400          320
11953           11959           107584          328
14081           14087           126736          356
17777           17783           160000          400
20353           20359           183184          428
22901           22907           206116          454
25813           25819           232324          482
26677           26683           240100          490
29813           29819           268324          518
32881           32887           295936          544
35093           35099           315844          562
51377           51383           462400          680
54133           54139           487204          698
76913           76919           692224          832
78773           78779           708964          842
82177           82183           739600          860
83713           83719           753424          868
104113          104119          937024          968
117877          117883          1060900         1030
122033          122039          1098304         1048
137393          137399          1236544         1112
162677          162683          1464100         1210
165377          165383          1488400         1220
170293          170299          1532644         1238
172501          172507          1552516         1246
180341          180347          1623076         1274
201301          201307          1811716         1346
209153          209159          1882384         1372
220273          220279          1982464         1408
229121          229127          2062096         1436
234901          234907          2114116         1454
246677          246683          2220100         1490

  Posted by Charlie on 2014-01-03 12:47:43
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