Puzzle 1211 906437281

Puzzle 1211 906437281

Gupta found that:

906437281 (prime), is the largest distinct digit prime obtained from sum of first n primes (case n=13306).

Q1. Find all the primes of this type (9 distinct digits)?

Q2. Find the all pandigitals (0-9 digits, composites) for the sum of the first n primes.

From March 7-14, 2025 contributions came from Giorgos Kalogeropoulos, Michael Branicky, Paul Cleary, Gennady Gusev, Simon Cavegn, Emmanuel Vantieghem

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Giorgos wrote:

Digits -> Primes -> Sum of n first primes

1 -> {2,5} -> {1,2}
2 -> {17,41} -> {4,6}
3 -> {197,281} -> {12,14}
4 -> {} -> {}
5 -> {28697,37561,38921,43201,61027,70241,86453} -> {108,122,124,130,152,162,178}
6 -> {132059,325019,329401,342761,681257} -> {216,326,328,334,458}
7 -> {1603597,2086159,3619807,4260917,4682791} -> {680,768,992,1070,1118}
8 -> {12598067,14782039,43109587,86340197} -> {1774,1912,3158,4378}
9 -> {906437281} -> {13306}
10 -> {} -> {}

Q2.{1063254978, 1829360475, 2835410967, 3478029561, 3954061782, 4271593608, 4583260179, 5190863247, 5741092638, 5897230146, 5932401786, 6980571324, 7803615924}.

These composite numbers are the sum of the first {14353, 18572, 22876, 25212, 26799, 27803, 28752, 30510, 32011, 32423, 32515, 35137, 37055} prime numbers respectively

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Michael wrote:

For Q1, I found no other such primes with exactly 9 digits.

For Q2, I found:

1063254978 (case n=14353) 1829360475 (case n=18572) 2835410967 (case n=22876) 3478029561 (case n=25212) 3954061782 (case n=26799) 4271593608 (case n=27803) 4583260179 (case n=28752) 5190863247 (case n=30510) 5741092638 (case n=32011) 5897230146 (case n=32423) 5932401786 (case n=32515) 6980571324 (case n=35137) 7803615924 (case n=37055)

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Paul wrote:

Q1. There are no more primes of that type.

Q2. Found 13 solutions.

Pandigital 1063254978 For the sum of first 14353 Primes.
Pandigital 1829360475 For the sum of first 18572 Primes.
Pandigital 2835410967 For the sum of first 22876 Primes.
Pandigital 3478029561 For the sum of first 25212 Primes.
Pandigital 3954061782 For the sum of first 26799 Primes.
Pandigital 4271593608 For the sum of first 27803 Primes.
Pandigital 4583260179 For the sum of first 28752 Primes.
Pandigital 5190863247 For the sum of first 30510 Primes.
Pandigital 5741092638 For the sum of first 32011 Primes.
Pandigital 5897230146 For the sum of first 32423 Primes.
Pandigital 5932401786 For the sum of first 32515 Primes.
Pandigital 6980571324 For the sum of first 35137 Primes.
Pandigital 7803615924 For the sum of first 37055 Primes.

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Gennady wrote:

Q1. No other numbers found

Q2. The list of numbers:

n, sum

14353, 1063254978

18572, 1829360475

22876, 2835410967

25212, 3478029561

26799, 3954061782

27803, 4271593608

28752, 4583260179

30510, 5190863247

32011, 5741092638

32423, 5897230146

32515, 5932401786

35137, 6980571324

37055, 7803615924

Addition to Q1.All prime sums with different digits:

n, sum

2, 5

4, 17

6, 41

12, 197

14, 281

108, 28697

122, 37561

124, 38921

130, 43201

152, 61027

162, 70241

178, 86453

216, 132059

326, 325019

328, 329401

334, 342761

458, 681257

680, 1603597

768, 2086159

992, 3619807

1070, 4260917

1118, 4682791

1774, 12598067

1912, 14782039

3158, 43109587

4378, 86340197

(13306, 906437281) - from puzzle.

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Simon wrote:

Q1 (I interpreted the question a tiny bit differently and found all the largest primes of this type up to base 17)

2 in base 2: '10' is prime, sum of first 1 primes.
5 in base 3: '12' is prime, sum of first 2 primes.
2 in base 4: '2' is prime, sum of first 1 primes.
17 in base 5: '32' is prime, sum of first 4 primes.
41 in base 6: '105' is prime, sum of first 6 primes.
197 in base 7: '401' is prime, sum of first 12 primes.
700897 in base 8: '2530741' is prime, sum of first 464 primes.
61027 in base 9: '102637' is prime, sum of first 152 primes.
906437281 in base 10: '906437281' is prime, sum of first 13306 primes.
24573774019 in base 11: 'a470289153' is prime, sum of first 64086 primes.
105803577787 in base 12: '186094a23b7' is prime, sum of first 128934 primes.
18741648173107 in base 13: 'a5c439870612' is prime, sum of first 1558946 primes.
787980284962853 in base 14: 'dc82640ba9713' is prime, sum of first 9528144 primes.
28313337946406791 in base 15: 'e8349d675c0ab1' is prime, sum of first 54297872 primes.
1146944040944361553 in base 16: 'feac38749bd6051' is prime, sum of first 329638366 primes.
48362364729869772209 in base 17: 'gf3e1bda68c50429' is prime, sum of first 2049398806 primes.

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Emmanuel wrote:

906437281 is the only one nine-digit prime that is the sum of the first primes.

Q2 : There are 13 pandigital composites that are the sum of the first n primes :
n
14353 1063254978 = 2*(3^3)*19689907
18572 1829360475 = (3^2)*(5^2)*(37^2)*5939
22876 2835410967 = (3^3)*59*61*29179
25212 3478029561 = (3^2)*263*379*3877
26799 3954061782 = 2*(3, 2)*11*29*688621
27803 4271593608 = (2^3)*(3^2)*59327689
28752 4583260179 = (3^6)*6287051
30510 5190863247 = (3^2)*59*9775637
32011 5741092638 = 2*(3^2)*11519*27689
32423 5897230146 = 2*(3^2)*19*379*45497
32515 5932401786 = 2*(3^2)*329577877
35137 6980571324 = (2^2)*(3^2)*29*43*131*1187
37055 7803615924 = (2^2)*(3^3)*13*101*113*487

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