## Number 64 – DigitSum() and DigitProduct()

Let   $DigitSum \,(N)$   be the sum of digits of   $N$,   and   $DigitProduct \,(N)$,
the product of digits of   $N$

To find a square number   $x$   such that

$DigitSum \,(x) - 1$     and     $DigitProduct \,(x) + 1$   are both squares

For example,     $x = 64 = 8^2$

$DigitSum \,(64) = 6 + 4 = 10$     and     $10 - 1 = 3^2$

$DigitProduct \,(64) = 6 \times 4 = 24$     and     $24 + 1 = 5^2$

Can you find another square like 64?

pipo found:

$187489 = 433^2$
$1+8+7+4+8+9 = 37$   ……….   $37 - 1 = 6^2$
$1\times 8\times 7\times 4\times 8\times 9 = 16128$   ….   $16128 + 1 = 127^2$

$529984 = 728^2$
$5+2+9+9+8+4 = 37$   ……….   $37 - 1 = 6^2$
$5\times 2\times 9\times 9\times 8\times 4 = 25920$   ….   $25920 + 1 = 161^2$

$2982529 = 1727^2$
$2+9+8+2+5+2+9 = 37$   ……….   $37 - 1 = 6^2$
$2\times 9\times 8\times 2\times 5\times 2\times 9 = 25920$   ….   $25920 + 1 = 161^2$

$6165289 = 2483^2$
$6+1+6+5+2+8+9 = 37$   ……….   $37 - 1 = 6^2$
$6\times 1\times 6\times 5\times 2\times 8\times 9 = 25920$   ….   $25920 + 1 = 161^2$

$45819361 = 6769^2$
$4+5+8+1+9+3+6+1 = 37$   ……….   $37 - 1 = 6^2$
$4\times 5\times 8\times 1\times 9\times 3\times 6\times 1 = 25920$   ….   $25920 + 1 = 161^2$

$55279225 = 7435^2$
$5+5+2+7+9+2+2+5 = 37$   ……….   $37 - 1 = 6^2$
$5\times 5\times 2\times 7\times 9\times 2\times 2\times 5 = 63000$   ….   $63000 + 1 = 251^2$

$59613841 = 7721^2$
$5+9+6+1+3+8+4+1 = 37$   ……….   $37 - 1 = 6^2$
$5\times 9\times 6\times 1\times 3\times 8\times 4\times 1 = 25920$   ….   $25920 + 1 = 161^2$

$85914361 = 9269^2$
$8+5+9+1+4+3+6+1 = 37$   ……….   $37 - 1 = 6^2$
$8\times 5\times 9\times 1\times 4\times 3\times 6\times 1 = 25920$   ….   $25920 + 1 = 161^2$

$89151364 = 9442^2$
$8+9+1+5+1+3+6+4 = 37$   ……….   $37 - 1 = 6^2$
$8\times 9\times 1\times 5\times 1\times 3\times 6\times 4 = 25920$   ….   $25920 + 1 = 161^2$

$114297481 = 10691^2$
$1+1+4+2+9+7+4+8+1 = 37$   ……….   $37 - 1 = 6^2$
$1\times 1\times 4\times 2\times 9\times 7\times 4\times 8\times 1 = 16128$   ….   $16128 + 1 = 127^2$

$118417924 = 10882^2$
$1+1+8+4+1+7+9+2+4 = 37$   ……….   $37 - 1 = 6^2$
$1\times 1\times 8\times 4\times 1\times 7\times 9\times 2\times 4 = 16128$   ….   $16128 + 1 = 127^2$

$181252369 = 13463^2$
$1+8+1+2+5+2+3+6+9 = 37$   ……….   $37 - 1 = 6^2$
$1\times 8\times 1\times 2\times 5\times 2\times 3\times 6\times 9 = 25920$   ….   $25920 + 1 = 161^2$

$183196225 = 13535^2$
$1+8+3+1+9+6+2+2+5 = 37$   ……….   $37 - 1 = 6^2$
$1\times 8\times 3\times 1\times 9\times 6\times 2\times 2\times 5 = 25920$   ….   $25920 + 1 = 161^2$

$223981156 = 14966^2$
$2+2+3+9+8+1+1+5+6 = 37$   ……….   $37 - 1 = 6^2$
$2\times 2\times 3\times 9\times 8\times 1\times 1\times 5\times 6 = 25920$   ….   $25920 + 1 = 161^2$

math grad - Interest: Number theory
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### 2 Responses to Number 64 – DigitSum() and DigitProduct()

1. paul says:

Here are those with x<=10000.

x = 46 … 10 – 1 = 3^2 & 24 + 1 = 5^2
x = 64 … 10 – 1 = 3^2 & 24 + 1 = 5^2
x = 113 … 5 – 1 = 2^2 & 3 + 1 = 2^2
x = 118 … 10 – 1 = 3^2 & 8 + 1 = 3^2
x = 131 … 5 – 1 = 2^2 & 3 + 1 = 2^2
x = 179 … 17 – 1 = 4^2 & 63 + 1 = 8^2
x = 181 … 10 – 1 = 3^2 & 8 + 1 = 3^2
x = 197 … 17 – 1 = 4^2 & 63 + 1 = 8^2
x = 226 … 10 – 1 = 3^2 & 24 + 1 = 5^2
x = 262 … 10 – 1 = 3^2 & 24 + 1 = 5^2
x = 311 … 5 – 1 = 2^2 & 3 + 1 = 2^2
x = 467 … 17 – 1 = 4^2 & 168 + 1 = 13^2
x = 476 … 17 – 1 = 4^2 & 168 + 1 = 13^2
x = 622 … 10 – 1 = 3^2 & 24 + 1 = 5^2
x = 647 … 17 – 1 = 4^2 & 168 + 1 = 13^2
x = 674 … 17 – 1 = 4^2 & 168 + 1 = 13^2
x = 719 … 17 – 1 = 4^2 & 63 + 1 = 8^2
x = 746 … 17 – 1 = 4^2 & 168 + 1 = 13^2
x = 764 … 17 – 1 = 4^2 & 168 + 1 = 13^2
x = 791 … 17 – 1 = 4^2 & 63 + 1 = 8^2
x = 811 … 10 – 1 = 3^2 & 8 + 1 = 3^2
x = 917 … 17 – 1 = 4^2 & 63 + 1 = 8^2
x = 971 … 17 – 1 = 4^2 & 63 + 1 = 8^2
x = 1135 … 10 – 1 = 3^2 & 15 + 1 = 4^2
x = 1153 … 10 – 1 = 3^2 & 15 + 1 = 4^2
x = 1234 … 10 – 1 = 3^2 & 24 + 1 = 5^2
x = 1243 … 10 – 1 = 3^2 & 24 + 1 = 5^2
x = 1315 … 10 – 1 = 3^2 & 15 + 1 = 4^2
x = 1324 … 10 – 1 = 3^2 & 24 + 1 = 5^2
x = 1342 … 10 – 1 = 3^2 & 24 + 1 = 5^2
x = 1351 … 10 – 1 = 3^2 & 15 + 1 = 4^2
x = 1358 … 17 – 1 = 4^2 & 120 + 1 = 11^2
x = 1385 … 17 – 1 = 4^2 & 120 + 1 = 11^2
x = 1423 … 10 – 1 = 3^2 & 24 + 1 = 5^2
x = 1432 … 10 – 1 = 3^2 & 24 + 1 = 5^2
x = 1513 … 10 – 1 = 3^2 & 15 + 1 = 4^2
x = 1531 … 10 – 1 = 3^2 & 15 + 1 = 4^2
x = 1538 … 17 – 1 = 4^2 & 120 + 1 = 11^2
x = 1583 … 17 – 1 = 4^2 & 120 + 1 = 11^2
x = 1835 … 17 – 1 = 4^2 & 120 + 1 = 11^2
x = 1853 … 17 – 1 = 4^2 & 120 + 1 = 11^2
x = 2134 … 10 – 1 = 3^2 & 24 + 1 = 5^2
x = 2143 … 10 – 1 = 3^2 & 24 + 1 = 5^2
x = 2267 … 17 – 1 = 4^2 & 168 + 1 = 13^2
x = 2276 … 17 – 1 = 4^2 & 168 + 1 = 13^2
x = 2314 … 10 – 1 = 3^2 & 24 + 1 = 5^2
x = 2341 … 10 – 1 = 3^2 & 24 + 1 = 5^2
x = 2413 … 10 – 1 = 3^2 & 24 + 1 = 5^2
x = 2431 … 10 – 1 = 3^2 & 24 + 1 = 5^2
x = 2447 … 17 – 1 = 4^2 & 224 + 1 = 15^2
x = 2474 … 17 – 1 = 4^2 & 224 + 1 = 15^2
x = 2627 … 17 – 1 = 4^2 & 168 + 1 = 13^2
x = 2672 … 17 – 1 = 4^2 & 168 + 1 = 13^2
x = 2726 … 17 – 1 = 4^2 & 168 + 1 = 13^2
x = 2744 … 17 – 1 = 4^2 & 224 + 1 = 15^2
x = 2762 … 17 – 1 = 4^2 & 168 + 1 = 13^2
x = 3115 … 10 – 1 = 3^2 & 15 + 1 = 4^2
x = 3124 … 10 – 1 = 3^2 & 24 + 1 = 5^2
x = 3142 … 10 – 1 = 3^2 & 24 + 1 = 5^2
x = 3151 … 10 – 1 = 3^2 & 15 + 1 = 4^2
x = 3158 … 17 – 1 = 4^2 & 120 + 1 = 11^2
x = 3185 … 17 – 1 = 4^2 & 120 + 1 = 11^2
x = 3214 … 10 – 1 = 3^2 & 24 + 1 = 5^2
x = 3241 … 10 – 1 = 3^2 & 24 + 1 = 5^2
x = 3412 … 10 – 1 = 3^2 & 24 + 1 = 5^2
x = 3421 … 10 – 1 = 3^2 & 24 + 1 = 5^2
x = 3446 … 17 – 1 = 4^2 & 288 + 1 = 17^2
x = 3464 … 17 – 1 = 4^2 & 288 + 1 = 17^2
x = 3511 … 10 – 1 = 3^2 & 15 + 1 = 4^2
x = 3518 … 17 – 1 = 4^2 & 120 + 1 = 11^2
x = 3581 … 17 – 1 = 4^2 & 120 + 1 = 11^2
x = 3644 … 17 – 1 = 4^2 & 288 + 1 = 17^2
x = 3815 … 17 – 1 = 4^2 & 120 + 1 = 11^2
x = 3851 … 17 – 1 = 4^2 & 120 + 1 = 11^2
x = 4123 … 10 – 1 = 3^2 & 24 + 1 = 5^2
x = 4132 … 10 – 1 = 3^2 & 24 + 1 = 5^2
x = 4213 … 10 – 1 = 3^2 & 24 + 1 = 5^2
x = 4231 … 10 – 1 = 3^2 & 24 + 1 = 5^2
x = 4247 … 17 – 1 = 4^2 & 224 + 1 = 15^2
x = 4274 … 17 – 1 = 4^2 & 224 + 1 = 15^2
x = 4312 … 10 – 1 = 3^2 & 24 + 1 = 5^2
x = 4321 … 10 – 1 = 3^2 & 24 + 1 = 5^2
x = 4346 … 17 – 1 = 4^2 & 288 + 1 = 17^2
x = 4364 … 17 – 1 = 4^2 & 288 + 1 = 17^2
x = 4427 … 17 – 1 = 4^2 & 224 + 1 = 15^2
x = 4436 … 17 – 1 = 4^2 & 288 + 1 = 17^2
x = 4463 … 17 – 1 = 4^2 & 288 + 1 = 17^2
x = 4472 … 17 – 1 = 4^2 & 224 + 1 = 15^2
x = 4634 … 17 – 1 = 4^2 & 288 + 1 = 17^2
x = 4643 … 17 – 1 = 4^2 & 288 + 1 = 17^2
x = 4724 … 17 – 1 = 4^2 & 224 + 1 = 15^2
x = 4742 … 17 – 1 = 4^2 & 224 + 1 = 15^2
x = 5113 … 10 – 1 = 3^2 & 15 + 1 = 4^2
x = 5131 … 10 – 1 = 3^2 & 15 + 1 = 4^2
x = 5138 … 17 – 1 = 4^2 & 120 + 1 = 11^2
x = 5183 … 17 – 1 = 4^2 & 120 + 1 = 11^2
x = 5311 … 10 – 1 = 3^2 & 15 + 1 = 4^2
x = 5318 … 17 – 1 = 4^2 & 120 + 1 = 11^2
x = 5381 … 17 – 1 = 4^2 & 120 + 1 = 11^2
x = 5678 … 26 – 1 = 5^2 & 1680 + 1 = 41^2
x = 5687 … 26 – 1 = 5^2 & 1680 + 1 = 41^2
x = 5768 … 26 – 1 = 5^2 & 1680 + 1 = 41^2
x = 5786 … 26 – 1 = 5^2 & 1680 + 1 = 41^2
x = 5813 … 17 – 1 = 4^2 & 120 + 1 = 11^2
x = 5831 … 17 – 1 = 4^2 & 120 + 1 = 11^2
x = 5867 … 26 – 1 = 5^2 & 1680 + 1 = 41^2
x = 5876 … 26 – 1 = 5^2 & 1680 + 1 = 41^2
x = 6227 … 17 – 1 = 4^2 & 168 + 1 = 13^2
x = 6272 … 17 – 1 = 4^2 & 168 + 1 = 13^2
x = 6344 … 17 – 1 = 4^2 & 288 + 1 = 17^2
x = 6434 … 17 – 1 = 4^2 & 288 + 1 = 17^2
x = 6443 … 17 – 1 = 4^2 & 288 + 1 = 17^2
x = 6578 … 26 – 1 = 5^2 & 1680 + 1 = 41^2
x = 6587 … 26 – 1 = 5^2 & 1680 + 1 = 41^2
x = 6722 … 17 – 1 = 4^2 & 168 + 1 = 13^2
x = 6758 … 26 – 1 = 5^2 & 1680 + 1 = 41^2
x = 6785 … 26 – 1 = 5^2 & 1680 + 1 = 41^2
x = 6857 … 26 – 1 = 5^2 & 1680 + 1 = 41^2
x = 6875 … 26 – 1 = 5^2 & 1680 + 1 = 41^2
x = 7226 … 17 – 1 = 4^2 & 168 + 1 = 13^2
x = 7244 … 17 – 1 = 4^2 & 224 + 1 = 15^2
x = 7262 … 17 – 1 = 4^2 & 168 + 1 = 13^2
x = 7424 … 17 – 1 = 4^2 & 224 + 1 = 15^2
x = 7442 … 17 – 1 = 4^2 & 224 + 1 = 15^2
x = 7568 … 26 – 1 = 5^2 & 1680 + 1 = 41^2
x = 7586 … 26 – 1 = 5^2 & 1680 + 1 = 41^2
x = 7622 … 17 – 1 = 4^2 & 168 + 1 = 13^2
x = 7658 … 26 – 1 = 5^2 & 1680 + 1 = 41^2
x = 7685 … 26 – 1 = 5^2 & 1680 + 1 = 41^2
x = 7856 … 26 – 1 = 5^2 & 1680 + 1 = 41^2
x = 7865 … 26 – 1 = 5^2 & 1680 + 1 = 41^2
x = 8135 … 17 – 1 = 4^2 & 120 + 1 = 11^2
x = 8153 … 17 – 1 = 4^2 & 120 + 1 = 11^2
x = 8315 … 17 – 1 = 4^2 & 120 + 1 = 11^2
x = 8351 … 17 – 1 = 4^2 & 120 + 1 = 11^2
x = 8513 … 17 – 1 = 4^2 & 120 + 1 = 11^2
x = 8531 … 17 – 1 = 4^2 & 120 + 1 = 11^2
x = 8567 … 26 – 1 = 5^2 & 1680 + 1 = 41^2
x = 8576 … 26 – 1 = 5^2 & 1680 + 1 = 41^2
x = 8657 … 26 – 1 = 5^2 & 1680 + 1 = 41^2
x = 8675 … 26 – 1 = 5^2 & 1680 + 1 = 41^2
x = 8756 … 26 – 1 = 5^2 & 1680 + 1 = 41^2
x = 8765 … 26 – 1 = 5^2 & 1680 + 1 = 41^2

Paul.

2. pipo says:

I looked at numbers, x, which are square, rest same conditions.
Format (x, x^2, productofdigits+1, square root of (pod+1), sum of digits -1, square root of (sod -1).
Strangely up till 1000000 all the square roots of (sod-1) are 6, except the first one.
8 64 25 5 9 3
433 187489 16129 127 36 6
728 529984 25921 161 36 6
1727 2982529 25921 161 36 6
2483 6165289 25921 161 36 6
6769 45819361 25921 161 36 6
7435 55279225 63001 251 36 6
7721 59613841 25921 161 36 6
9269 85914361 25921 161 36 6
9442 89151364 25921 161 36 6
10691 114297481 16129 127 36 6
10882 118417924 16129 127 36 6
13463 181252369 25921 161 36 6
13535 183196225 25921 161 36 6
14966 223981156 25921 161 36 6
15227 231861529 25921 161 36 6
20278 411197284 16129 127 36 6
21131 446519161 25921 161 36 6
24139 582691321 25921 161 36 6
24479 599221441 25921 161 36 6
25381 644195161 25921 161 36 6
29611 876811321 16129 127 36 6
30689 941814721 16129 127 36 6
34127 1164652129 25921 161 36 6
35531 1262451961 25921 161 36 6
38161 1456261921 25921 161 36 6
40211 1616924521 25921 161 36 6
40807 1665211249 25921 161 36 6
42911 1841353921 25921 161 36 6
47161 2224159921 25921 161 36 6
50129 2512916641 25921 161 36 6
52111 2715556321 63001 251 36 6
57815 3342574225 201601 449 36 6
59489 3538941121 25921 161 36 6
74312 5522273344 201601 449 36 6
90073 8113145329 25921 161 36 6
97579 9521661241 25921 161 36 6
105434 11116328356 25921 161 36 6
105569 11144813761 16129 127 36 6
111419 12414193561 25921 161 36 6
118792 14111539264 25921 161 36 6
127331 16213183561 25921 161 36 6
134623 18123352129 25921 161 36 6
138979 19315162441 25921 161 36 6
146087 21341411569 25921 161 36 6
152704 23318511616 25921 161 36 6
154315 23813119225 25921 161 36 6
178415 31831912225 25921 161 36 6
270521 73181611441 16129 127 36 6
279611 78182311321 16129 127 36 6
334846 112121843716 16129 127 36 6
337987 114235212169 25921 161 36 6
339796 115461321616 25921 161 36 6
343682 118117317124 9409 97 36 6
355123 126112345129 25921 161 36 6
363142 131872112164 16129 127 36 6
368261 135616164121 25921 161 36 6
380944 145118331136 25921 161 36 6
383239 146872131121 16129 127 36 6
388927 151264211329 25921 161 36 6
391339 153146212921 25921 161 36 6
426869 182217143161 16129 127 36 6
462511 213916425121 25921 161 36 6
558046 311415338116 25921 161 36 6
561572 315363111184 25921 161 36 6
567046 321541166116 25921 161 36 6
575515 331217515225 63001 251 36 6
584954 342171182116 16129 127 36 6
721882 521113621924 25921 161 36 6
788761 622143915121 25921 161 36 6
901189 812141613721 16129 127 36 6

pipo