## Prime number – sum of the digits

Find a prime number such that

>>   the sum of the digits of its square is a square, and
>>   the square of the prime number is also a sum of five consecutive prime numbers square of a prime number is a sum of five consecutive prime numbers.

math grad - Interest: Number theory
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### 2 Responses to Prime number – sum of the digits

1. pipo says:

31 is the smallest prime that fits the rules: 31^2 = 961, 961 is the sum of five consecutive primes (181+191+193+197+199= 961) and the sum of the digits of 961 is 16 or sod(961) = 4^2
And four more:
41: 41^2 =1681 (=317+331+337+347+349) and sod(1681) = 4^2
283: 283^2 =80089 (=15991+16001+16007+16033+16057) and sod(80089) = 5^2
1201: 1201^2 = 1442401 (=288461+288467+288481+288493+288499) and sod(1442401) = 4^2
1427: 1427^2 = 2036329 (=407249+407257+407263+407273+407287) and sod(2036329) = 5^2.

pipo

2. paul says:

and a few more

12101^2 = 146434201 = 29286853 + 29286823 + 29286791 + 29286857 + 29286877
36761^2 = 1351371121 = 270274231 + 270274223 + 270274187 + 270274237 + 270274243
53917^2 = 2907042889 = 581408587 + 581408557 + 581408543 + 581408599 + 581408603
60887^2 = 3707226769 = 741445357 + 741445339 + 741445337 + 741445363 + 741445373
66841^2 = 4467719281 = 893543867 + 893543809 + 893543797 + 893543881 + 893543927
71881^2 = 5166878161 = 1033375633 + 1033375601 + 1033375573 + 1033375657 + 1033375697
81097^2 = 6576723409 = 1315344703 + 1315344649 + 1315344623 + 1315344713 + 1315344721
88793^2 = 7884196849 = 1576839373 + 1576839361 + 1576839287 + 1576839401 + 1576839427
90173^2 = 8131169929 = 1626233993 + 1626233971 + 1626233941 + 1626234007 + 1626234017
101977^2 = 10399308529 = 2079861709 + 2079861691 + 2079861689 + 2079861713 + 2079861727

Paul.