## Prime numbers p – DigitSum(p) = DigitProduct(p)

Find prime numbers for which the sum of digits equals the product of digits

For example,

4-digit prime numbers using digits:    $(1, 1, 2, 4)$

$2141$
$2411$
$4211$

$1\times 1\times 2\times 4 \; = \; 8 \; = \; 1 + 1 + 2 + 4$

5-digit prime numbers using digits:    $(1, 1, 1, 2, 5)$

$11251$
$12511$
$15121$
$25111$

$1\times 1\times 1\times 2\times 5 \; = \; 10 \; = \; 1 + 1 + 1 + 2 + 5$

5-digit prime numbers using digits:    $(1, 1, 2, 2, 2)$

$21221$

$1\times 1\times 2\times 2\times 2 \; = \; 8 \; = \; 1 + 1 + 2 + 2 + 2$

7-digit prime numbers using digits:    $(1, 1, 1, 1, 1, 2, 7)$

$1112171$
$1127111$
$1172111$
$1271111$
$7112111$

$1\times 1\times 1\times 1\times 1\times 2\times 7 \; = \; 14 \; = \; 1 + 1 + 1 + 1 + 1 + 2 + 7$

8-digit prime numbers using digits:    $(1, 1, 1, 1, 1, 1, 2, 8)$

$11112811$
$11128111$

$1\times 1\times 1\times 1\times 1\times 1\times 2\times 8 \; = \; 16 \; = \; 1 + 1 + 1 + 1 + 1 + 1 + 2 + 8$

Any 6-digit primes?

Find other 8-digit prime numbers sharing this property.

Find 9-digit primes, and larger primes.

math grad - Interest: Number theory
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### 3 Responses to Prime numbers p – DigitSum(p) = DigitProduct(p)

1. David @InfinitelyManic says:

I’m testing some code to resolve a rounding error issue so there could be false positives…

p, digitSum, digitProd
104021 8 8
105211 10 10
110251 10 10
112501 10 10
115021 10 10
115201 10 10
120041 8 8
120401 8 8
120511 10 10
121501 10 10
122021 8 8
122201 8 8
124001 8 8
125101 10 10
150211 10 10
151201 10 10
201151 10 10
201401 8 8
201511 10 10
202121 8 8
204101 8 8
205111 10 10
210401 8 8
211051 10 10
211501 10 10
221021 8 8
221201 8 8
222011 8 8
240011 8 8
240101 8 8
401201 8 8
412001 8 8
501121 10 10
510121 10 10
511201 10 10
512011 10 10
512101 10 10
520111 10 10

2. David @InfinitelyManic says:

Some 8 digit primes
I’m testing some code to resolve a rounding error issue so there could be false positives…
p, digitSum, digitProd

10001521 10 10
10004201 8 8
10010251 10 10
10012501 10 10
10020041 8 8
10020151 10 10
10020401 8 8
10021051 10 10
10025011 10 10
10025101 10 10
10042001 8 8
10051021 10 10
10102501 10 10
10112171 14 14
10121117 14 14
10125001 10 10
10150201 10 10
10171121 14 14
10171211 14 14
10202021 8 8
10205011 10 10
10205101 10 10
10211171 14 14
10211711 14 14
10215001 10 10
10222001 8 8
10400021 8 8
10420001 8 8
10501201 10 10
10502101 10 10
10520101 10 10
10521001 10 10
11002051 10 10
11017211 14 14
11020501 10 10
11027111 14 14
11050021 10 10
11050201 10 10
11101217 14 14
11110217 14 14
11110271 14 14
11112811 16 16
11117021 14 14
11120171 14 14
11128111 16 16
….

3. David @InfinitelyManic says:

Some 9 digit primes

I’m testing some code to resolve a rounding error issue so there could be false positives…
p, digitSum, digitProd

100000421 8 8
100001204 8 8
100002212 8 8
100005121 10 10
100012116 12 12
100016211 12 12
100020003 6 6
100020122 8 8
100020212 8 8
100020511 10 10
100021105 10 10
100022201 8 8
100040201 8 8
100041002 8 8
100042001 8 8
100050211 10 10
100061121 12 12
100102004 8 8
100102105 10 10
100102116 12 12
100102202 8 8
100106121 12 12
100110052 10 10
100111127 14 14
100111721 14 14
100112171 14 14
100116201 12 12
100117211 14 14
100120116 12 12
100120611 12 12
100121711 14 14
100141131 12 12
100160211 12 12
100171121 14 14
100201105 10 10
100210116 12 12
100211117 14 14
100250011 10 10