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Problem 700

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Eulercoin

Leonhard Euler was born on $15$ April $1707$.

Consider the sequence $1504170715041707n \bmod 4503599627370517$.

An element of this sequence is defined to be an Eulercoin if it is strictly smaller than all previously found Eulercoins.

For example, the first term is $1504170715041707$ which is the first Eulercoin. The second term is $3008341430083414$ which is greater than $1504170715041707$ so is not an Eulercoin. However, the third term is $8912517754604$ which is small enough to be a new Eulercoin.

The sum of the first $2$ Eulercoins is therefore $1513083232796311$.

Find the sum of all Eulercoins.

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欧拉币

莱昂哈德·欧拉出生于$1707$年四月$15$日。

考虑序列$1504170715041707n \bmod 4503599627370517$。

该序列中的某个元素被定义为欧拉币，当且仅当它严格小于之前已经定义的所有欧拉币。

例如，序列的第一项是$1504170715041707$，这是第一枚欧拉币。序列的第二项是$3008341430083414$，因为它大于$1504170715041707$，因此它不是欧拉币。然后，序列的第三项是$8912517754604$，比第一项要小，因此是一枚新的欧拉币。

因此，前$2$枚欧拉币之和为$1513083232796311$。

求所有欧拉币之和。

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